_ ,0 ^ , xdx (1 + x)(1 + x^2 ) = - Vidyadip

MCQ

$\int_{\,0}^{\,\infty } {\frac{{xdx}}{{(1 + x)(1 + {x^2})}} = } $

A
$0$
B
$\pi /2$
✓
$\pi /4$
D
$1$

✓

Answer

Correct option: C.

$\pi /4$

c
(c) $\int_0^\infty {\frac{{xdx}}{{(1 + x)(1 + {x^2})}} = \int_0^\infty {\frac{{ - \frac{1}{2}dx}}{{(1 + x)}} + \int_0^\infty {\frac{{\left( {\frac{1}{2}x + \frac{1}{2}} \right)}}{{1 + {x^2}}}dx} } } $

$ = \left[ {\frac{{ - 1}}{2}\log (1 + x)} \right]_0^\infty + \frac{1}{2} \times \frac{1}{2}[\log \,(1 + {x^2})]\,_0^\infty + \frac{1}{2}[{\tan ^{ - 1}}x]\,_0^\infty $

$ = 0 + 0 + \frac{1}{2}\left[ {\frac{\pi }{2} - 0} \right] = \frac{\pi }{4}$.

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

Choose the correct answers from the given four options:
If $\text{f(x)}=\begin{cases}\text{mx}+1,&\text{if x}\leq\frac{\pi}{2}\\\sin\text{x}+\text{n},&\text{if x}>\frac{\pi}{2}\end{cases},$ is continuous at $\text{x}=\frac{\pi}{2},$ then:

The set of all values of $k$ for which $\left(\tan ^{-1} x \right)^{3}+\left(\cot ^{-1} x \right)^{3}= k \pi^{3}, x \in R$, is the interval

Choose the correct answer from the given four options:
The curve $\text{y}=\text{x}^{\frac{1}{5}}$ has at (0, 0)

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

The slope of the tangent to the curve $x=3 t^2+1, y=t^3-1$ at $x=1$ is:

The maximum solution of the objective function lies :

Let R be a relation on N defined by x + 2y = 8. The domain of R is:

Let $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ be three unit vectors, such that $\big|\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big|=1$ and $\vec{\text{a}}$ is perpendicular to $\vec{\text{b}}.$ If $\vec{\text{c}}$ makes angle $\alpha$ and $\beta$ with $\vec{\text{a}}$ and $\vec{\text{b}}$ respectively, then $\cos\alpha+\cos\beta=$

If $x^y=e^{x-y}$ then $\frac{\text{dy}}{\text{dx}}$ is :

The function $f(x) = x^x$ decreases on the interval :