Download App

Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals1 Mark
MCQ
Evaluate: $\int_0^1 \frac{x \tan ^{-1} x}{\left(1+x^2\right)^{3 / 2}} d x$
  • A
    $\frac{4-\pi}{2 \sqrt{2}}$
  • B
    $\frac{4+\pi}{2 \sqrt{2}}$
  • ✓
    $\frac{4-\pi}{4 \sqrt{2}}$
  • D
    None of these
✓

Answer

Correct option: C.
$\frac{4-\pi}{4 \sqrt{2}}$
(c): Let $\int_0^1 \frac{x \tan ^{-1} x}{\left(1+x^2\right)^{3 / 2}} d x$ Put $\tan ^{-1} x=\theta \Rightarrow x=\tan \theta \Rightarrow d x=\sec ^2 \theta d \theta$ When, $x=0 \Rightarrow \theta=0$ and $x=1 \Rightarrow \theta=\frac{\pi}{4}$ $I=\int_0^1 \frac{x \tan ^{-1} x}{\left(1+x^2\right)^{3 / 2}} d x=\int_0^{\pi / 4} \frac{\theta \tan \theta}{\sec ^3 \theta} \sec ^2 \theta d \theta$$=\int_0^{\pi / 4} \theta \sin \theta d \theta=[-\theta \cos \theta]_0^{\pi / 4}-\int_0^{\pi / 4}(-\cos \theta) d \theta$[Integrating by parts]
$=[-\theta \cos \theta]_0^{\pi / 4}+[\sin \theta]_0^{\pi / 4}=\frac{4-\pi}{4 \sqrt{2}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Integrals→Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $A$ be a matrix such that $A.\,\left[ {\begin{array}{*{20}{c}}
1&2\\
0&3
\end{array}} \right]$ is a scalar matrix and $\left| {3A} \right| = 108$ . Then $A^2$ equals
→
The vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ satisfy the equation $2\vec{\text{a}}+\vec{\text{b}}=\vec{\text{p}}$ and $\vec{\text{a}}+2\vec{\text{b}}=\vec{\text{q}},$ where $\vec{\text{p}}=\hat{\text{i}}+\hat{\text{j}}$ and $\vec{\text{q}}=\hat{\text{i}}-\hat{\text{j}}.$ If $\theta$ is the angle between $\vec{\text{a}}$ and $\vec{\text{b}},$ then :
→
$\int_0^2 {\frac{{{x^3}\,dx}}{{{{({x^2} + 1)}^{\frac{3}{2}}}}}} = $
→
$\int\text{e}^{\text{x}}\Big(\frac{1-\sin\text{x}}{1-\cos\text{x}}\Big)\text{dx}=$
→
$I=\int \limits_{\pi / 4}^{\pi / 3}\left(\frac{8 \sin x-\sin 2 x}{x}\right) d x$. Then
→
The equation of tangent to the curve $y(1 + x^2) = 2 - x, w$ here it crosses $x-$axis is:
→
If m[-3 amp; 4] + n[-3 amp; 4] = [10 amp;-11], then 3m + 7n =
→
Choose the correct answer from the given four options.The value of $\cos^{-1}\Big(\cos\frac{3\pi}{2}\Big)$ is equal to:
→
If area bounded by the curves ${y^2} = 4\,ax$ and $y = mx$ is ${a^2}/3,$, then the value of $m$ is
→
At $\text{x}=\frac{5\pi}{6}, $ $\text{f}(\text{x})=2\sin3\text{x}+3 \cos3\text{x}$ is:
→