Q.Evaluate the definite integral: $\int_{0}^{1} x e^{x^2} \, dx$

The integral $\int \sqrt{x^2 + 2x} \, dx$ involves the square root of a quadratic expression. Integrals of the form $\int \sqrt{ax^2 + bx + c} \, dx$ are typically solved by completing the square for the quadratic expression $ax^2 + bx + c$. This transforms the integral into one of three standard forms: $\int \sqrt{a^2 - u^2} \, du$, $\int \sqrt{u^2 + a^2} \, du$, or $\int \sqrt{u^2 - a^2} \, du$, for which direct formulas exist.

In this problem, we will complete the square for $x^2 + 2x$ and then apply the appropriate standard integral formula.

1. Complete the square for the expression under the square root: The quadratic expression is $x^2 + 2x$. To complete the square, we add and subtract the square of half the coefficient of $x$. The coefficient of $x$ is $2$, so half of it is $1$, and its square is $1^2 = 1$.

Now the integral becomes $\int \sqrt{(x+1)^2 - 1^2} \, dx$.

2. Identify the standard integral form:

Let $u = x+1$. Then $du = dx$. The integral transforms to $\int \sqrt{u^2 - 1^2} \, du$.

This is of the form $\int \sqrt{u^2 - a^2} \, du$, where $a=1$.

3. Apply the standard integral formula: The standard formula for $\int \sqrt{y^2 - a^2} \, dy$ is:

   $\int \sqrt{y^2 - a^2} \, dy = \frac{y}{2}\sqrt{y^2 - a^2} - \frac{a^2}{2}\cosh^{-1}\left(\frac{y}{a}\right) + C$

   Applying this formula with $y=u$ and $a=1$:

> [!WARNING]
> Be careful with the sign and the inverse hyperbolic function. For $\sqrt{y^2 - a^2}$, it's $\cosh^{-1}$ with a negative sign. For $\sqrt{y^2 + a^2}$, it's $\sinh^{-1}$ with a positive sign. …

The key here is that the denominator has a power of $3/2$, which is an awkward exponent. Direct substitution or partial fractions won't help. But if we can rewrite the expression so that the square root in the denominator becomes a simple trigonometric function, the integration becomes straightforward.

The form $x^2 - a^2$ under a square root (or a power of it) is a classic signal for the substitution $x = a \sec \theta$. Why? Because $\sec^2 \theta - 1 = \tan^2 \theta$, so $x^2 - a^2 = a^2 \tan^2 \theta$, and the square root becomes $a |\tan \theta|$. For $x > a$ (the usual domain), $\tan \theta > 0$, so we can drop the absolute value.

Let's work through it.

1. Substitute $x = a \sec \theta$.

   Then $dx = a \sec \theta \tan \theta \, d\theta$.

   Also, $x^2 - a^2 = a^2 (\sec^2 \theta - 1) = a^2 \tan^2 \theta$.

   Therefore $(x^2 - a^2)^{3/2} = (a^2 \tan^2 \theta)^{3/2} = a^3 |\tan \theta|^3$. For $\theta \in (0, \pi/2)$ (so $x > a$), $\tan \theta > 0$, so this is $a^3 \tan^3 \theta$.

2. Rewrite the integral.

3. Simplify the trigonometric expression. $\frac{\sec \theta}{\tan^2 \theta} = \frac{1/\cos \theta}{\sin^2 \theta / \cos^2 \theta} = \frac{1}{\cos \theta} \cdot \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\cos \theta}{\sin^2 \theta} = \cot \theta \csc \theta$. So the integral becomes

4. Integrate. Recall that $\frac{d}{d\theta} (\csc \theta) = -\cot \theta \csc \theta$. Hence $\int \cot \theta \csc \theta \, d\theta = -\csc \theta + C$. So

5. Back-substitute to $x$. From $x = a \sec \theta$, we have $\sec \theta = \frac{x}{a}$. …

We are asked to evaluate

The presence of $\sec x$ and $\tan x$ together suggests using the Weierstrass substitution $t = \tan\frac{x}{2}$. This substitution turns trigonometric integrals into rational ones, which we can handle with partial fractions. The trick is to express $\sec x$ and $\tan x$ in terms of $t$, simplify, and then integrate.

1. Recall the standard Weierstrass substitution formulas:

Also, $dx = \frac{2}{1+t^2} \, dt$.

2. Rewrite the integrand in terms of $t$: The denominator is

Notice $1 + t^2 + 2t = (1+t)^2$, so

Expand:

So denominator becomes $\frac{t^2 + 6t + 5}{1-t^2}$.

The numerator $\sec x$ is $\frac{1+t^2}{1-t^2}$, and $dx = \frac{2}{1+t^2} dt$.

Hence the integral becomes

The factors $(1+t^2)$ and $(1-t^2)$ cancel neatly, leaving

3. Factor the quadratic and use partial fractions: $t^2 + 6t + 5 = (t+1)(t+5)$. So

Multiply through: $2 = A(t+5) + B(t+1)$.

Set $t = -1$: $2 = A(4) \Rightarrow A = \frac{1}{2}$. …

The standard form of this integral splits into an inverse-hyperbolic-secant part and an arcsine part. Writing the integrand as

and integrating each term:

since $\dfrac{d}{dx}\operatorname{Sech}^{-1}\sqrt{x}=-\dfrac{1}{2x\sqrt{1-x}}$ and $\dfrac{d}{dx}\sin^{-1}\sqrt{x}=\dfrac{1}{2\sqrt{x}\sqrt{1-x}}$.

Comparing with the given form $2f(x)-2\sin^{-1}\sqrt{x}+c$, the non-arcsine part is the inverse-hyperbolic-secant term, so …

Concept. A half-angle trigonometric substitution rationalises $\sqrt{\dfrac{1-\sqrt{x}}{1+\sqrt{x}}}$, because $\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}} = \tan\dfrac{\theta}{2}$.

Step 1 — substitute. Let $\sqrt{x}=\cos\theta$, $\theta\in(0,\tfrac{\pi}{2})$, so $x=\cos^2\theta$, $dx=-2\cos\theta\sin\theta\,d\theta$:

Step 2 — simplify with half-angles. $\tan\dfrac{\theta}{2}\,\sin\theta = 2\sin^2\dfrac{\theta}{2} = 1-\cos\theta$, hence

Step 3 — return to $x$. With $\theta=\cos^{-1}\sqrt{x}$: $2\theta = \pi - 2\sin^{-1}\sqrt{x}$ (the $\pi$ is absorbed into the constant), $\sec\theta+\tan\theta = \dfrac{1+\sqrt{1-x}}{\sqrt{x}}$. So

Step 4 — identify $f$. Comparing with $I = 2f(x)-2\sin^{-1}\sqrt{x}+c$: …

The core idea here is that two different antiderivatives of the same function can differ by a constant. When you evaluate an indefinite integral using two different substitutions, you get expressions that look different but are actually the same up to an additive constant. The question asks for the difference between those two expressions, including their arbitrary constants — and that difference is just a constant.

Let’s work through both methods carefully.

1. First method: $x = \tan \theta$ Substitute $x = \tan \theta$, so $dx = \sec^2 \theta \, d\theta$ and $1 + x^2 = 1 + \tan^2 \theta = \sec^2 \theta$. The integral becomes:

Let $u = \sin \theta$, then $du = \cos \theta \, d\theta$, giving:

Since $\sin \theta = \frac{x}{\sqrt{1+x^2}}$, we get:

2. Second method: $x^2 + 1 = Z$ Substitute $Z = x^2 + 1$, so $dZ = 2x \, dx$ and $x^2 = Z - 1$. The integral becomes:

Since $x \, dx = \frac{dZ}{2}$ and $x^2 = Z-1$, we have:

Integrating:

Substitute back $Z = 1 + x^2$:

3. Compare the two results Write $f(x) + K$ from method 1:

Write $g(x) + C$ from method 2:

Now compute $f(x) - g(x) + K - C$:

1. Simplify the denominator. Using $\cos^2 x=1-\sin^2 x$,

2. Substitute $t=\sin x$, so $dt=\cos x\,dx$:

3. Reduce to standard form. Since $25-16t^2=25\!\left(1-\big(\tfrac{4t}{5}\big)^2\right)$,

Let $u=\tfrac{4t}{5}$, $dt=\tfrac54\,du$:

$\frac15\cdot\frac54\int\frac{du}{\sqrt{1-u^2}}=\frac14\sin^{-1}u+c=\frac14\sin^{-1}\!\left(\frac{4t}{5}\right)+c.$ …

The concept: recognise the answer, then differentiate to confirm

When an integral like this appears with the answer's shape already given, the fastest and safest route is to differentiate the proposed antiderivative and see whether it reproduces the integrand. The key algebraic hint is the split of the angle:

which is exactly the kind of decomposition the compound-angle formula

is built for.

Step 1 — Differentiate the candidate

Let

By the product rule (and the chain rule on $\sin^{100}x$):

Step 2 — Factor out $100\sin^{99}x$

The bracket is precisely $\sin(100x + x)$:

Step 3 — Integrate

Dividing by 100 and integrating back: …

The key insight is that the expression $\frac{1+\sqrt{3}\cot x}{1-\sqrt{3}\cot x}$ looks like a tangent addition formula in disguise. Recall that $\cot x = \frac{\cos x}{\sin x}$, and $\sqrt{3} = \tan(\pi/3)$. This suggests rewriting the integrand in terms of $\tan$ or $\sin$ and $\cos$ to reveal a simpler structure.

1. Rewrite in terms of sine and cosine Since $\cot x = \frac{\cos x}{\sin x}$, we have:

2. Recognize a tangent addition formula Notice that $\sin x + \sqrt{3}\cos x$ and $\sin x - \sqrt{3}\cos x$ resemble the expansion of $\sin(x \pm \pi/3)$ because:

and similarly,

Multiplying numerator and denominator by 2, we get:

3. Use a trigonometric identity to simplify further The ratio of sines can be expressed using the identity:

Here $A = x + \pi/3$, $B = x - \pi/3$, so $A-B = 2\pi/3$. Thus:

Since $\cos(2\pi/3) = -1/2$ and $\sin(2\pi/3) = \sqrt{3}/2$, we have:

4. Integrate term by term …