Q.Integrate the following function: $\frac{2x}{1+x^2}$

Showing the 12 most recent of 16 on this concept.

1. COMEDK 2026Set 2026-M1 markMCQ
   Q.

   [!FORMULA] $\int \frac{e^{\log \left(1+\frac{1}{x^2}\right)}}{x^2+\frac{1}{x^2}} d x=$

   (A) $\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2+1}{x \sqrt{2}}\right)+C$ (B) $\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{x^2-1}{\sqrt{2} x}\right)+C$ (C) $-\frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C$ (D) $\frac{1}{\sqrt{2}} \tan ^{-1}\left(x-\frac{1}{x}\right)+C$

   ›Reveal solutionSolution

   The integrand simplifies dramatically using exponent rules and algebraic manipulation, leading to a standard arctangent integral; the correct antiderivative matches option (D).

   We start with the integral

   $\int \frac{e^{\log \left(1+\frac{1}{x^2}\right)}}{x^2+\frac{1}{x^2}} \, dx.$

   Concept & Intuition

   The presence of $e^{\log(\cdots)}$ is a huge clue: for any positive argument, $e^{\log(u)} = u$. That immediately collapses the numerator into something algebraic. Then the denominator is symmetric in $x$ and $1/x$, which often suggests a substitution like $t = x - 1/x$ because its derivative appears in the numerator. This is a classic trick for integrals involving $x^2 + 1/x^2$.

   ---

   Step-by-step solution

   1. Simplify the exponential Since $e^{\log(u)} = u$ for $u>0$ (and $1+1/x^2 > 0$ for all real $x \neq 0$), we have

   $e^{\log\left(1+\frac{1}{x^2}\right)} = 1 + \frac{1}{x^2}.$

   So the integral becomes

   $\int \frac{1 + \frac{1}{x^2}}{x^2 + \frac{1}{x^2}} \, dx.$

   2. Rewrite numerator and denominator Multiply numerator and denominator by $x^2$ to clear fractions:

   $\frac{1 + \frac{1}{x^2}}{x^2 + \frac{1}{x^2}} = \frac{x^2 + 1}{x^4 + 1}.$

   So the integral is

   $\int \frac{x^2 + 1}{x^4 + 1} \, dx.$

   3. Divide numerator and denominator by $x^2$ This is the key algebraic trick:

   $\frac{x^2 + 1}{x^4 + 1} = \frac{1 + \frac{1}{x^2}}{x^2 + \frac{1}{x^2}}.$

   Notice that the numerator $1 + 1/x^2$ is the derivative of $x - 1/x$ (since $\frac{d}{dx}(x - 1/x) = 1 + 1/x^2$).

   Also, $x^2 + 1/x^2 = (x - 1/x)^2 + 2$.

   4. Substitute Let $t = x - \frac{1}{x}$. Then

   $dt = \left(1 + \frac{1}{x^2}\right) dx.$

   And

   $x^2 + \frac{1}{x^2} = t^2 + 2.$

   The integral becomes

   $\int \frac{dt}{t^2 + 2}.$

   5. Integrate This is a standard arctangent form:

   $\int \frac{dt}{t^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{t}{a}\right) + C.$

   Here $a = \sqrt{2}$, so

   $\int \frac{dt}{t^2 + 2} = \frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{t}{\sqrt{2}}\right) + C.$

   6. Back-substitute Replace $t$ with $x - \frac{1}{x}$:

   $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x - \frac{1}{x}}{\sqrt{2}}\right) + C.$

   Simplify the argument:

   $\frac{x - \frac{1}{x}}{\sqrt{2}} = \frac{x^2 - 1}{\sqrt{2}\, x}.$

   So the antiderivative is

   $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x^2 - 1}{\sqrt{2}\, x}\right) + C.$

   Tip

   Notice that option (B) has $\frac{x^2-1}{\sqrt{2}x}$ inside the arctan, but with a minus sign in front? Actually (B) is $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x^2-1}{\sqrt{2}x}\right)+C$ — that matches exactly! But wait, check (D): $\frac{1}{\sqrt{2}} \tan^{-1}\left(x-\frac{1}{x}\right)+C$. Are these the same?

   No: $\tan^{-1}(x-1/x)$ is not equal to $\tan^{-1}((x^2-1)/(\sqrt{2}x))$ in general. However, our result has the $\sqrt{2}$ inside the arctan argument. Let’s re-check: we got $\frac{1}{\sqrt{2}} \tan^{-1}\left(\frac{x-1/x}{\sqrt{2}}\right)$. That is not the same as $\frac{1}{\sqrt{2}} \tan^{-1}(x-1/x)$. So (D) is missing the division by $\sqrt{2}$ inside. But (B) has exactly $\frac{x^2-1}{\sqrt{2}x}$ which is $\frac{x-1/x}{\sqrt{2}}$. So (B) matches our result.

   Watch out

   A common mistake is to forget the factor $1/\sqrt{2}$ inside the arctan argument. Option (D) tempts you by dropping it, but that would give a different derivative. Always check by differentiating.

   Thus the correct choice is (B).

   ✓Final answer

   The correct option is (B).

   ANSWER: B

2. COMEDK 2025Set 2025-A1 markMCQ
   Q.$\int \frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)}\left(1+x+x^2\right) d x=$ (A) $e^{\tan ^{-1} x}+c$ (B) $x e^{\tan ^{-1} x}+c$ (C) $\frac{e^{\tan ^{-1} x}}{\left(1+x^2\right)}+c$ (D) $\frac{x e^{\tan ^{-1} x}}{\left(1+x^2\right)}+c$

   ›Reveal solutionSolution

   The integral simplifies by substituting $u = \tan^{-1} x$, which turns the expression into a sum of a standard exponential integral and a derivative-of-product pattern, yielding $x e^{\tan^{-1} x} + C$. The correct option is (B).

   ---

   The key insight is that the denominator $1+x^2$ is exactly the derivative of $\tan^{-1} x$, so the substitution $u = \tan^{-1} x$ is natural. Once we do that, the polynomial $1+x+x^2$ becomes something in terms of $\tan u$, and we can split the integral into two recognizable pieces.

   1. Substitute $u = \tan^{-1} x$. Then $du = \frac{dx}{1+x^2}$, and $x = \tan u$. The integral becomes

   $\int e^{\tan^{-1} x} \cdot \frac{1+x+x^2}{1+x^2}\,dx = \int e^{u} \left(1 + \tan u + \tan^2 u\right) du.$

   2. Simplify the trigonometric expression. Recall $1 + \tan^2 u = \sec^2 u$. So

   $1 + \tan u + \tan^2 u = \sec^2 u + \tan u.$

   The integral is now

   $\int e^{u} (\sec^2 u + \tan u)\, du.$

   3. Split and recognize patterns.

   $\int e^{u} \sec^2 u \, du + \int e^{u} \tan u \, du.$

   Notice that $\frac{d}{du}(\tan u) = \sec^2 u$. The first integral is of the form $\int e^{u} f'(u) du$ with $f(u) = \tan u$, and the second is $\int e^{u} f(u) du$.

   4. Use the product rule in reverse. For any differentiable $f(u)$,

   $\frac{d}{du}\left( e^{u} f(u) \right) = e^{u} f'(u) + e^{u} f(u).$

   Here $f(u) = \tan u$, so

   $\frac{d}{du}\left( e^{u} \tan u \right) = e^{u} \sec^2 u + e^{u} \tan u.$

   That is exactly our integrand. Therefore,

   $\int e^{u} (\sec^2 u + \tan u)\, du = e^{u} \tan u + C.$

   5. Back-substitute $u = \tan^{-1} x$. Since $\tan(\tan^{-1} x) = x$, we get

   $e^{\tan^{-1} x} \cdot x + C = x e^{\tan^{-1} x} + C.$

   Watch out

   A common mistake is to try integrating by parts directly without the substitution, or to forget that $1 + \tan^2 u = \sec^2 u$, missing the neat cancellation.

   Tip

   The pattern $\int e^u (f'(u) + f(u))\,du = e^u f(u) + C$ is a powerful shortcut — it’s just the product rule in disguise.

   ✓Final answer

   The correct option is (B).

   ANSWER: B

3. COMEDK 2023Set 2023-M1 markMCQ
   Q.$\int \frac{x d x}{2(1+x)^{3 / 2}}$ is equal to (A) $\frac{2+x}{\sqrt{1+x}}+C$ (B) $\frac{2+x}{x \sqrt{1+x}}+C$ (C) $\frac{x}{\sqrt{1+x}}+C$ (D) $-\frac{x}{\sqrt{1+x}}+C$

   ›Reveal solutionSolution

   With $u=1+x$, the integral becomes $\tfrac12\int(u^{-1/2}-u^{-3/2})\,du=u^{1/2}+u^{-1/2}=\dfrac{2+x}{\sqrt{1+x}}+C$.

   $\displaystyle\int\frac{x\,dx}{2(1+x)^{3/2}}.$ Let $u=1+x\Rightarrow x=u-1,\ dx=du$:

   $\frac12\int\frac{u-1}{u^{3/2}}\,du=\frac12\int\left(u^{-1/2}-u^{-3/2}\right)du.$

   $=\frac12\left(2u^{1/2}+2u^{-1/2}\right)=u^{1/2}+u^{-1/2}=\sqrt{1+x}+\frac{1}{\sqrt{1+x}}.$

   Combine over a common denominator:

   $\frac{(1+x)+1}{\sqrt{1+x}}=\frac{2+x}{\sqrt{1+x}}+C.$

   ✓Final answer

   The correct option is (A) — $\frac{2+x}{\sqrt{1+x}}+C$

4. COMEDK 2021Set 20211 markMCQ
   Q.$\int {{{{2^x}} \over {\sqrt {1 - {4^x}} }}dx}$ is equal to (A) $(\log 2){\sin ^{ - 1}}{2^x} + C$ (B) ${1 \over 2}{\sin ^{ - 1}}{2^x} + C$ (C) ${1 \over {\log 2}}{\sin ^{ - 1}}{2^x} + C$ (D) $2\log 2{\sin ^{ - 1}}{2^x} + C$

   ›Reveal solutionSolution

   I = (1/log 2) * integral du / sqrt(1 - u^2) = (1/log 2) * arcsin(u) + C = (1/log 2) * arcsin(2^x) + C.

   Concept: substitution reducing the integrand to the standard form 1/sqrt(1 - u^2), whose integral is arcsin(u).

   I = integral 2^x / sqrt(1 - 4^x) dx. Note 4^x = (2^x)^2.

   Put u = 2^x. Then du = 2^x * log 2 dx, so 2^x dx = du / log 2.

   I = (1/log 2) * integral du / sqrt(1 - u^2)

   = (1/log 2) * arcsin(u) + C

   = (1/log 2) * arcsin(2^x) + C.

   ✓Final answer

   The correct option is (C) — ${1 \over {\log 2}}{\sin ^{ - 1}}{2^x} + C$

   ANSWER: C

5. KCET 2023Set A-21 markMCQ
   Q.If $u = \sin^{-1}\left(\dfrac{2x}{1+x^2}\right)$ and $v = \tan^{-1}\left(\dfrac{2x}{1-x^2}\right)$ then $\dfrac{du}{dv}$ is (A) $2$ (B) $\dfrac{1-x^2}{1+x^2}$ (C) $1$ (D) $\dfrac{1}{2}$

   ›Reveal solutionSolution

   The key idea is to simplify $u$ and $v$ using standard inverse trigonometric identities before differentiating. The result is $\frac{du}{dv} = 1$.

   We are given two functions of $x$:

   $u = \sin^{-1}\left(\frac{2x}{1+x^2}\right), \quad v = \tan^{-1}\left(\frac{2x}{1-x^2}\right).$

   The question asks for $\frac{du}{dv}$, the derivative of $u$ with respect to $v$. A direct approach — differentiating each with respect to $x$ and then dividing — is possible, but messy. A cleaner path is to recognize that both expressions are standard forms for inverse trigonometric functions when $x = \tan \theta$.

   1. Substitute $x = \tan \theta$. This is a classic trick because $\frac{2x}{1+x^2}$ and $\frac{2x}{1-x^2}$ appear in double-angle formulas. Let $\theta = \tan^{-1} x$, so $x = \tan \theta$. Then:

   $\frac{2x}{1+x^2} = \frac{2\tan\theta}{1+\tan^2\theta} = \sin 2\theta.$

   Hence,

   $u = \sin^{-1}(\sin 2\theta).$

   2. Simplify $u$ carefully. The identity $\sin^{-1}(\sin \alpha) = \alpha$ holds only when $\alpha$ is in the principal range $[-\pi/2, \pi/2]$. Here $\alpha = 2\theta$, and $\theta = \tan^{-1} x$ lies in $(-\pi/2, \pi/2)$. So $2\theta$ lies in $(-\pi, \pi)$. For $u$ to be defined as the principal value of $\sin^{-1}$, we need $2\theta \in [-\pi/2, \pi/2]$, which corresponds to $x \in [-1, 1]$. In that interval, we can safely write:

   $u = 2\theta = 2\tan^{-1} x.$

   Watch out

   Outside $x \in [-1, 1]$, the simplification $u = 2\tan^{-1} x$ would be off by a constant (like $\pi - 2\tan^{-1} x$). However, the derivative $\frac{du}{dx}$ remains the same because the constant vanishes. So the derivative result holds for all $x$ where the functions are defined.

   3. Simplify $v$ similarly. Using $x = \tan \theta$ again:

   $\frac{2x}{1-x^2} = \frac{2\tan\theta}{1-\tan^2\theta} = \tan 2\theta.$

   Therefore,

   $v = \tan^{-1}(\tan 2\theta).$

   The principal value of $\tan^{-1}(\tan \alpha)$ equals $\alpha$ when $\alpha \in (-\pi/2, \pi/2)$. Here $2\theta$ lies in $(-\pi, \pi)$, so for $2\theta \in (-\pi/2, \pi/2)$ — again $x \in (-1, 1)$ — we have:

   $v = 2\theta = 2\tan^{-1} x.$

   Tip

   Both $u$ and $v$ simplify to the same expression $2\tan^{-1} x$ on the interval $(-1, 1)$. That immediately tells us $u = v$ (up to a constant, which is zero here), so their derivatives with respect to each other are 1.

   4. Differentiate to find $\frac{du}{dv}$. Since $u = 2\tan^{-1} x$ and $v = 2\tan^{-1} x$, we have $u = v$ for all $x$ in the domain where both simplifications hold. Differentiating with respect to $v$:

   $\frac{du}{dv} = 1.$

   Even if we consider the general case (including $x$ outside $(-1,1)$), the derivatives $\frac{du}{dx}$ and $\frac{dv}{dx}$ are both $\frac{2}{1+x^2}$, so their ratio is still 1.

   ✓Final answer

   The value is $\boxed{1}$, which corresponds to option (C).

6. COMEDK 2026Set 2026-M1 markMCQ
   Q.

   [!FORMULA] $\int \frac{x+1}{x\left(1+x e^x\right)} d x=$

   (A) $\log \left|c x e^x\left(1+x e^x\right)\right|$ (B) $\log \left|c x e^x\left(1+x e^x\right)\right|$ (C) $\log \left|\frac{c\left(1+x e^x\right)}{x e^x}\right|$ (D) $\log \left|\frac{c x e^x}{1+x e^x}\right|$

   ›Reveal solutionSolution

   The integral simplifies by noticing the derivative of $x e^x$ appears in the denominator; the result is $\log\left|\frac{c\,x e^x}{1+x e^x}\right|$, which matches option (D).

   ---

   The key insight is that the integrand contains $x e^x$ in the denominator, and the derivative of $x e^x$ is $e^x(1+x)$. That derivative is almost exactly the numerator $x+1$, except for a factor of $e^x$. This suggests a substitution or a clever split of the fraction to reveal a logarithmic derivative.

   ---

   1. Rewrite the integrand to expose the derivative of $x e^x$. Notice that

   $\frac{d}{dx}(x e^x) = e^x + x e^x = e^x(1+x).$

   Our numerator is $x+1$, so we can write:

   $\frac{x+1}{x(1+x e^x)} = \frac{e^x(x+1)}{e^x \cdot x(1+x e^x)} = \frac{e^x(1+x)}{x e^x (1+x e^x)}.$

   The numerator is now exactly the derivative of $x e^x$.

   2. Perform a substitution. Let $t = x e^x$. Then $dt = e^x(1+x)\,dx$. The integral becomes:

   $\int \frac{e^x(1+x)}{x e^x (1+x e^x)}\,dx = \int \frac{dt}{t(1+t)}.$

   3. Decompose the rational function. Use partial fractions:

   $\frac{1}{t(1+t)} = \frac{1}{t} - \frac{1}{1+t}.$

   So the integral is:

   $\int \left(\frac{1}{t} - \frac{1}{1+t}\right) dt = \log|t| - \log|1+t| + C = \log\left|\frac{t}{1+t}\right| + C.$

   4. Substitute back. Since $t = x e^x$, we have:

   $\int \frac{x+1}{x(1+x e^x)}\,dx = \log\left|\frac{x e^x}{1 + x e^x}\right| + C.$

   The constant $C$ can be written as $\log|c|$ to combine logs:

   $= \log\left|\frac{c\,x e^x}{1 + x e^x}\right|.$

   Tip

   The trick is spotting that $x e^x$ is a natural "inner function" because its derivative appears in the numerator after multiplying by $e^x$. This is a classic pattern: whenever you see $x e^x$ and $x+1$ together, think of the derivative of $x e^x$.

   Watch out

   A common mistake is to try splitting the fraction as $\frac{x+1}{x} \cdot \frac{1}{1+x e^x}$ and then integrating by parts — that leads nowhere. The substitution $t = x e^x$ is the clean path.

   ✓Final answer

   The correct option is (D).

   ANSWER: D

7. COMEDK 2024Set 2024-M1 markMCQ
   Q.

   [!FORMULA] $\text { The value of } \int \frac{1}{x+\sqrt{x-1}} d x \text { is }$

   (A) $\log (x+\sqrt{x-1})+\sin ^{-1}\left(\sqrt{\frac{x-1}{x}}\right)+C$ (B) $\log (x+\sqrt{x-1})-\frac{2}{\sqrt{3}} \tan ^{-1}\left(\frac{2 \sqrt{x-1}+1}{\sqrt{3}}\right)+C$ (C) $\log (x+\sqrt{x-1})+C$ (D) $\log (x-1+\sqrt{x-1})+\frac{1}{\sqrt{3}} \log \left|\frac{x-2-\sqrt{3}}{x-2+\sqrt{3}}\right|+C$

   ›Reveal solutionSolution

   The integral simplifies by substituting $t = \sqrt{x-1}$, turning it into a rational function that integrates to a logarithm and an arctangent, matching option (B).

   We are asked to evaluate

   $\int \frac{1}{x + \sqrt{x-1}} \, dx.$

   The presence of $\sqrt{x-1}$ suggests a substitution that removes the square root, turning the integrand into a rational function. The trick is to set $t = \sqrt{x-1}$, so that $x = t^2 + 1$ and $dx = 2t \, dt$. This transforms the integral into a form we can handle with partial fractions or a standard arctangent formula.

   Let’s work through it step by step.

   1. Substitute $t = \sqrt{x-1}$. Then $x = t^2 + 1$ and $dx = 2t \, dt$. The denominator becomes

   $x + \sqrt{x-1} = (t^2 + 1) + t = t^2 + t + 1.$

   So the integral becomes

   $\int \frac{1}{t^2 + t + 1} \cdot 2t \, dt = 2 \int \frac{t}{t^2 + t + 1} \, dt.$

   2. Prepare for integration by rewriting the numerator to match the derivative of the denominator. The derivative of $t^2 + t + 1$ is $2t + 1$. We have $2t$ in the numerator, so write

   $2t = (2t + 1) - 1.$

   Then

   $2 \int \frac{t}{t^2 + t + 1} \, dt = \int \frac{2t + 1}{t^2 + t + 1} \, dt - \int \frac{1}{t^2 + t + 1} \, dt.$

   3. First integral:

   $\int \frac{2t + 1}{t^2 + t + 1} \, dt = \log|t^2 + t + 1| + C_1.$

   Since $t^2 + t + 1 > 0$ for all real $t$, we can drop the absolute value.

   4. Second integral: Complete the square in the denominator:

   $t^2 + t + 1 = \left(t + \frac12\right)^2 + \frac34.$

   So

   $\int \frac{1}{t^2 + t + 1} \, dt = \int \frac{1}{\left(t + \frac12\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \, dt.$

   Using the formula $\int \frac{du}{u^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right)$, with $u = t + \frac12$ and $a = \frac{\sqrt{3}}{2}$, we get

   $\int \frac{1}{t^2 + t + 1} \, dt = \frac{2}{\sqrt{3}} \tan^{-1}\left( \frac{2t + 1}{\sqrt{3}} \right) + C_2.$

   5. Combine results:

   $2 \int \frac{t}{t^2 + t + 1} \, dt = \log(t^2 + t + 1) - \frac{2}{\sqrt{3}} \tan^{-1}\left( \frac{2t + 1}{\sqrt{3}} \right) + C.$

   6. Back-substitute $t = \sqrt{x-1}$:

   $t^2 + t + 1 = (x-1) + \sqrt{x-1} + 1 = x + \sqrt{x-1}.$

   Hence

   $\int \frac{1}{x + \sqrt{x-1}} \, dx = \log(x + \sqrt{x-1}) - \frac{2}{\sqrt{3}} \tan^{-1}\left( \frac{2\sqrt{x-1} + 1}{\sqrt{3}} \right) + C.$

   This matches option (B) exactly.

   Watch out

   A common mistake is to try a direct substitution like $u = x + \sqrt{x-1}$, but that leads to a messy derivative. The substitution $t = \sqrt{x-1}$ is cleaner because it eliminates the square root entirely.

   Tip

   Notice that the logarithm term $\log(x + \sqrt{x-1})$ appears in multiple options, so the distinguishing feature is the arctangent term with $\frac{2}{\sqrt{3}}$. That alone points to option (B).

   ✓Final answer

   The correct option is (B).

   ANSWER: B

8. COMEDK 2021Set 2021-B1 markMCQ
   Q.If $\displaystyle\int\dfrac{2^x}{\sqrt{1-4^x}}\,dx=k\sin^{-1}(2^x)+c$, then k is (A) $\dfrac{\log 2}{2}$ (B) $\dfrac{1}{\log 2}$ (C) $\dfrac{1}{2\log 2}$ (D) $\log 2$

   ›Reveal solutionSolution

   $k=\dfrac{1}{\log 2}$.

   Let $u=2^x$, so $du=2^x\ln2\,dx$ and $4^x=u^2$. Then

   $\int\frac{2^x}{\sqrt{1-4^x}}\,dx=\int\frac{u}{\sqrt{1-u^2}}\cdot\frac{du}{u\ln2}=\frac{1}{\ln2}\int\frac{du}{\sqrt{1-u^2}}=\frac{1}{\ln2}\sin^{-1}u+c.$

   So the integral is $\frac{1}{\log2}\sin^{-1}(2^x)+c$, giving $k=\dfrac{1}{\log 2}$.

   ✓Final answer

   The correct option is (B) — $\dfrac{1}{\log 2}$

9. COMEDK 2026Set 2026-A1 markMCQ
   Q.

   [!FORMULA] $\int \frac{d x}{x \sqrt{x^2+4}}=$

   (A) $\frac{1}{4} \log \left|\frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}\right|+C$ (B) $\frac{1}{4} \log \left|\frac{\sqrt{x^2+4}+2}{\sqrt{x^2+4}-2}\right|+C$ (C) $\frac{1}{2} \log \left|\frac{\sqrt{x^2+4}+2}{\sqrt{x^2+4}-2}\right|+C$ (D) $\frac{1}{2} \log \left|\frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}\right|+C$

   ›Reveal solutionSolution

   Substituting $x=2\tan\theta$ gives $\frac{1}{4}\log\left|\frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}\right|+C$ — option (A).

   For $\displaystyle\int\frac{dx}{x\sqrt{x^2+4}}$ put $x=2\tan\theta$, so $dx=2\sec^2\theta\,d\theta$ and $\sqrt{x^2+4}=2\sec\theta$:

   $\int\frac{2\sec^2\theta\,d\theta}{2\tan\theta\cdot 2\sec\theta}=\frac{1}{2}\int\csc\theta\,d\theta=\frac{1}{2}\log\left|\csc\theta-\cot\theta\right|+C$

   With $\csc\theta=\dfrac{\sqrt{x^2+4}}{x}$ and $\cot\theta=\dfrac{2}{x}$:

   $=\frac{1}{2}\log\left|\frac{\sqrt{x^2+4}-2}{x}\right|+C$

   Since $\dfrac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}=\dfrac{(\sqrt{x^2+4}-2)^2}{x^2}=\left(\dfrac{\sqrt{x^2+4}-2}{x}\right)^2$, we can write

   $\frac{1}{2}\log\left|\frac{\sqrt{x^2+4}-2}{x}\right|=\frac{1}{4}\log\left|\frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}\right|$

   ✓Final answer

   $\displaystyle\int\frac{dx}{x\sqrt{x^2+4}}=\frac{1}{4}\log\left|\frac{\sqrt{x^2+4}-2}{\sqrt{x^2+4}+2}\right|+C$ — option (A).

10. COMEDK 2024Set 2024-M1 markMCQ
    Q.If $\int \frac{1}{\sqrt{\sin ^3 x \cos x}} d x=\frac{k}{\sqrt{\tan x}}+c$ then the value of $k$ is (A) $-2$ (B) 1 (C) 2 (D) $-1$

    ›Reveal solutionSolution

    The integral simplifies by rewriting the integrand in terms of $\tan x$, leading to a straightforward power rule integration; comparing the result with the given form shows $k = -2$.

    We are given

    $\int \frac{1}{\sqrt{\sin^3 x \cos x}}\, dx = \frac{k}{\sqrt{\tan x}} + c$

    and need to find $k$.

    Concept and intuition

    The integrand mixes powers of $\sin x$ and $\cos x$. A classic trick is to express everything in terms of $\tan x$ (or $\cot x$) because the derivative of $\tan x$ is $\sec^2 x$, which itself is $1/\cos^2 x$. This often turns messy trigonometric integrals into simple power rules. Here, the presence of $\sqrt{\tan x}$ on the right side is a strong hint: the integrand likely simplifies to something like $(\tan x)^{-3/2} \cdot \sec^2 x$, whose antiderivative is a constant times $(\tan x)^{-1/2}$.

    Let’s work it out step by step.

    1. Rewrite the integrand using $\tan x$.

    $\frac{1}{\sqrt{\sin^3 x \cos x}} = \frac{1}{\sin^{3/2} x \cdot \cos^{1/2} x}$

    Divide numerator and denominator by $\cos^{3/2} x$ (a common trick to introduce $\tan x$):

    $= \frac{1}{\cos^{3/2} x \cdot \tan^{3/2} x \cdot \cos^{1/2} x} = \frac{1}{\cos^{2} x \cdot \tan^{3/2} x}$

    because $\cos^{3/2} x \cdot \cos^{1/2} x = \cos^2 x$.

    Since $1/\cos^2 x = \sec^2 x$, we have:

    $\frac{1}{\sqrt{\sin^3 x \cos x}} = \frac{\sec^2 x}{\tan^{3/2} x}.$

    2. Set up the substitution. Let $u = \tan x$. Then $du = \sec^2 x \, dx$. The integral becomes:

    $\int \frac{\sec^2 x}{\tan^{3/2} x}\, dx = \int \frac{du}{u^{3/2}} = \int u^{-3/2}\, du.$

    3. Integrate using the power rule.

    $\int u^{-3/2}\, du = \frac{u^{-3/2+1}}{-3/2+1} = \frac{u^{-1/2}}{-1/2} = -2\, u^{-1/2} + C.$

    4. Substitute back. Since $u = \tan x$, we get:

    $\int \frac{1}{\sqrt{\sin^3 x \cos x}}\, dx = -2\, (\tan x)^{-1/2} + C = -\frac{2}{\sqrt{\tan x}} + C.$

    5. Compare with the given form. The problem states the integral equals $\frac{k}{\sqrt{\tan x}} + c$. Matching coefficients, we see $k = -2$.

    Watch out

    A common mistake is to forget the negative sign from the power rule: $\int u^{-3/2} du = -2 u^{-1/2}$, not $+2 u^{-1/2}$. Always check the exponent carefully.

    Tip

    The substitution $u = \tan x$ is powerful whenever the integrand is a product of powers of $\sin x$ and $\cos x$ — just aim to express everything as $\sec^2 x$ times a power of $\tan x$.

    ✓Final answer

    The correct option is (A).

    ANSWER: A

11. COMEDK 2024Set 2024-A1 markMCQ
    Q.

    [!FORMULA] $\int \frac{f^{\prime}(x)}{f(x) \log (f(x))} d x \text { is equal to }$

    (A) $f(x) \log f(x)+C$ (B) $\frac{1}{\log (\log f(x))}+C$ (C) $\frac{f(x)}{\log f(x)}+C$ (D) $\log (\log f(x))+C$

    ›Reveal solutionSolution

    The integral simplifies by substitution $u = \log(f(x))$, leading directly to $\log(\log f(x)) + C$, so the correct choice is (D).

    Concept & Intuition

    When you see a fraction where the numerator is the derivative of the denominator’s “inside,” a substitution is almost always the cleanest path. Here, the denominator is $f(x) \log(f(x))$, and the numerator is $f'(x)$. Notice that the derivative of $\log(f(x))$ is $\frac{f'(x)}{f(x)}$, which appears in the integrand. That suggests setting $u = \log(f(x))$, turning the whole integral into a simple $\int \frac{du}{u}$.

    Step-by-step solution

    1. Identify the substitution Let $u = \log(f(x))$. Then differentiate:

    $\frac{du}{dx} = \frac{f'(x)}{f(x)} \quad\Rightarrow\quad du = \frac{f'(x)}{f(x)}\,dx.$

    2. Rewrite the integral The original integral is

    $\int \frac{f'(x)}{f(x) \log(f(x))}\,dx.$

    Factor the $f(x)$ in the denominator:

    $\int \frac{f'(x)/f(x)}{\log(f(x))}\,dx.$

    Now substitute $u = \log(f(x))$ and $du = \frac{f'(x)}{f(x)}dx$:

    $\int \frac{du}{u}.$

    3. Integrate The integral $\int \frac{du}{u}$ is a standard result:

    $\int \frac{du}{u} = \log|u| + C.$

    4. Back-substitute Replace $u$ with $\log(f(x))$:

    $\log|\log(f(x))| + C.$

    Since $f(x)$ is presumably positive (otherwise $\log(f(x))$ wouldn’t be defined in real numbers), we can drop the absolute value:

    $\log(\log f(x)) + C.$

    5. Match with options This matches option (D) exactly.

    Watch out

    A common mistake is to try integration by parts or to misidentify the derivative of $\log(f(x))$ as $\frac{1}{f(x)}$ instead of $\frac{f'(x)}{f(x)}$. Always check the chain rule carefully.

    Tip

    If you ever see $\frac{f'(x)}{f(x)}$ in an integrand, think “logarithmic derivative.” That pattern almost always signals a substitution $u = \log f(x)$ or $u = f(x)$.

    ✓Final answer

    The correct option is (D).

    ANSWER: D

12. COMEDK 2023Set 2023-M1 markMCQ
    Q.$\int \frac{4^x}{\sqrt{1-16^x}} d x$ is equal to (A) $(\log 4) \sin ^{-1} 4^x+C$ (B) $\frac{1}{4} \sin ^{-1}\left(4^x\right)+C$ (C) $\frac{1}{\log 4} \sin ^{-1} 4^x+C$ (D) $4 \log 4 \sin ^{-1} 4+C$

    ›Reveal solutionSolution

    Put $u=4^x$ so $16^x=u^2$ and $du=4^x\ln4\,dx$; the integral becomes $\dfrac{1}{\ln4}\int\dfrac{du}{\sqrt{1-u^2}}=\dfrac{1}{\log4}\sin^{-1}(4^x)+C$.

    $\displaystyle\int\frac{4^x}{\sqrt{1-16^x}}\,dx.$ Let $u=4^x\Rightarrow du=4^x\ln4\,dx\Rightarrow 4^x\,dx=\dfrac{du}{\ln4}$, and $16^x=(4^x)^2=u^2$:

    $\int\frac{1}{\sqrt{1-u^2}}\cdot\frac{du}{\ln4}=\frac{1}{\ln4}\sin^{-1}u+C=\frac{1}{\log4}\sin^{-1}(4^x)+C.$

    ✓Final answer

    The correct option is (C) — $\frac{1}{\log 4}\sin^{-1}4^{x}+C$