Evaluate $\displaystyle\int \frac{dx}{x^2+4x+13}$.

Evaluate ∫ (dx)/(x²+4x+13).

Example · Example 8

\displaystyle\int \frac{dx}{x^2+4x+13}

West Bengal WbchseTextbookSubjectiveImportance★★★★★est

17% · 11/65 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Complete the square: $x^2+4x+13=(x+2)^2+9=(x+2)^2+3^2$ . With $u=x+2$ , this is the standard

form $\int du/(u^2+3^2)$ from Section 5:

$\int\frac{dx}{x^2+4x+13} = \int\frac{du}{u^2+9} = \frac13\tan^{-1}\!\left(\frac u3\right)+C = \frac13\tan^{-1}\!\left(\frac{x+2}{3}\right)+C.$ …

Start your 14-day free trial to unlock the full solution →

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

Showing the 12 most recent of 30 on this concept.

CBSE 2026Set ANNUAL1 markMCQ
$Q. \int (from 0 to 1) dx/\sqrt(1-x²) is equal to:$
(a) π/2
(b) π/4
(c) π/3
(d) π/6
›Reveal solutionSolution

$\frac{1}{\sqrt{1-x^2}}$ is the derivative of $\sin^{-1}x$ , so the definite integral is just $\sin^{-1}(1) - \sin^{-1}(0)$ .

We know the standard result $\displaystyle\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}x + C$ .

So: …

Start your 14-day free trial to see the full solution →
CBSE 2025Set ANNUAL1 markMCQ
$\displaystyle\int_{\sqrt2}^{x} \dfrac{dt}{t\sqrt{t^2-1}} = \dfrac{\pi}{12}$
(a) 2
(b) 1
(c) 0
(d) -1
›Reveal solutionSolution

The antiderivative of $\dfrac{1}{t\sqrt{t^2-1}}$ is $\sec^{-1}t$ ; apply the limits and solve for $x$ .

We know the standard result:

$\int\dfrac{dt}{t\sqrt{t^2-1}} = \sec^{-1}|t| + C$

So

$\int_{\sqrt2}^{x}\dfrac{dt}{t\sqrt{t^2-1}} = \sec^{-1}x - \sec^{-1}\sqrt2$

Since $\sec\dfrac{\pi}{4} = \sqrt2$ , we have $\sec^{-1}\sqrt2 = \dfrac{\pi}{4}$ .

Given this equals $\dfrac{\pi}{12}$ :

$\sec^{-1}x - \dfrac{\pi}{4} = \dfrac{\pi}{12}$ …

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \sqrt{x^2-a^2}\,dx$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\sqrt{x^2-a^2}\,dx$ matches entry (e), the standard NCERT formula for this integral.

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \sqrt{a^2-x^2}\,dx$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\sqrt{a^2-x^2}\,dx$ matches entry (c), the standard NCERT formula for this integral.

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \dfrac{dx}{\sqrt{x^2-a^2}}$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\dfrac{dx}{\sqrt{x^2-a^2}}$ matches entry (d).

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \dfrac{dx}{\sqrt{a^2-x^2}}$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\dfrac{dx}{\sqrt{a^2-x^2}}$ matches entry (b).

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \dfrac{dx}{x^2-a^2}$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\dfrac{dx}{x^2-a^2}$ matches entry (a).

…

Start your 14-day free trial to see the full solution →
CBSE 2024Set A1 mark
$\int \dfrac{dx}{a^2-x^2}$
(a) $\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+c$
(b) $\sin^{-1}\dfrac{x}{a}+c$
(c) $\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{x}{a}+c$
(d) $\log\left|x+\sqrt{x^2-a^2}\right|+c$
(e) $\dfrac{x}{2}\sqrt{x^2-a^2}-\dfrac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+c$
(f) $\sec^{-1}x$
(g) $\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+c$
(h) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a}+c$
(i) $\tan^{-1}x$ . Which entry of Column B matches this Column A item?
›Reveal solutionSolution

$\displaystyle\int\dfrac{dx}{a^2-x^2}$ matches entry (g).

…

Start your 14-day free trial to see the full solution →
CBSE 2023Set A1 mark
$\int \tan x\,dx$
(a) $\log|\sin x| + c$
(b) $\cos 2x$
(c) $\sec^2 x$
(d) $2\cos 2x$
(e) $\log|\text{cosec}\,x - \cot x| + c$
(f) $-\log|\cos x| + c$
(g) $\log|\sec x + \tan x| + c$
›Reveal solutionSolution

$\displaystyle\int\tan x\,dx=-\log|\cos x|+c$ , matching option (f).

Write $\tan x=\dfrac{\sin x}{\cos x}$ and substitute $u=\cos x$ , $du=-\sin x\,dx$ : …

Start your 14-day free trial to see the full solution →
CBSE 2023Set A1 mark
$\int \cot x\,dx$
(a) $\log|\sin x| + c$
(b) $\cos 2x$
(c) $\sec^2 x$
(d) $2\cos 2x$
(e) $\log|\text{cosec}\,x - \cot x| + c$
(f) $-\log|\cos x| + c$
(g) $\log|\sec x + \tan x| + c$
›Reveal solutionSolution

$\displaystyle\int\cot x\,dx=\log|\sin x|+c$ , matching option (a).

Write $\cot x=\dfrac{\cos x}{\sin x}$ and substitute $u=\sin x$ , $du=\cos x\,dx$ : …

Start your 14-day free trial to see the full solution →
CBSE 2023Set A1 mark
$\int \sec x\,dx$
(a) $\log|\sin x| + c$
(b) $\cos 2x$
(c) $\sec^2 x$
(d) $2\cos 2x$
(e) $\log|\text{cosec}\,x - \cot x| + c$
(f) $-\log|\cos x| + c$
(g) $\log|\sec x + \tan x| + c$
›Reveal solutionSolution

$\displaystyle\int\sec x\,dx=\log|\sec x+\tan x|+c$ , matching option (g). This is a standard result — a memorised integral, verified by differentiating the RHS.

…

Start your 14-day free trial to see the full solution →
CBSE 2023Set A1 mark
$\int \text{cosec}\,x\,dx$
(a) $\log|\sin x| + c$
(b) $\cos 2x$
(c) $\sec^2 x$
(d) $2\cos 2x$
(e) $\log|\text{cosec}\,x - \cot x| + c$
(f) $-\log|\cos x| + c$
(g) $\log|\sec x + \tan x| + c$
›Reveal solutionSolution

$\displaystyle\int \text{cosec}\,x\,dx=\log|\text{cosec}\,x-\cot x|+c$ , matching option (e). A standard integral, verified by differentiating the RHS.

…

Start your 14-day free trial to see the full solution →

←Previous questionExample 7 — Evaluate ∫ (dx)/(x²-9).→Next questionExample 9 — Evaluate ∫ (2x+3)/(x²+4x+13)dx.

Q.Evaluate $\displaystyle\int \frac{dx}{x^2+4x+13}$ .

Concept understanding — Standard Integral Forms

Q.Evaluate ∫dxx2+4x+13\displaystyle\int \frac{dx}{x^2+4x+13}∫x2+4x+13dx​.

🎓Unlock everything free for 14 days

Q.Evaluate $\displaystyle\int \frac{dx}{x^2+4x+13}$ .

Unlock everything free for 14 days