_ ^ x^2 ^ - 1 x^3 1 + x^6 ;dx is equal to - Vidyadip

MCQ

$\int_{}^{} {\frac{{{x^2}{{\tan }^{ - 1}}{x^3}}}{{1 + {x^6}}}\;dx} $ is equal to

A
${\tan ^{ - 1}}({x^3}) + c$
✓
$\frac{1}{6}{({\tan ^{ - 1}}{x^3})^2} + c$
C
$ - \frac{1}{2}{({\tan ^{ - 1}}{x^3})^2} + c$
D
$\frac{1}{2}{({\tan ^{ - 1}}{x^2})^3} + c$

✓

Answer

Correct option: B.

$\frac{1}{6}{({\tan ^{ - 1}}{x^3})^2} + c$

b
(b) Put ${x^3} = t \Rightarrow dt = 3{x^2}\,dx$
$\int_{}^{} {\frac{{{x^2}{{\tan }^{ - 1}}{x^3}\,dx}}{{1 + {x^6}}}} = \frac{1}{3}\int_{}^{} {\frac{{{{\tan }^{ - 1}}t}}{{1 + {t^2}}}} \,dt$
Put $z = {\tan ^{ - 1}}t \Rightarrow dz = \frac{{dt}}{{1 + {t^2}}}$
$ = \frac{1}{3}\int_{}^{} {z\,dz} = \frac{1}{3}\frac{{{z^2}}}{2} = \frac{{{z^2}}}{6} = \frac{1}{6}{({\tan ^{ - 1}}{x^3})^2} + c$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Consider two sets A and B , each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and q respectively. Let d and D be the common differences of AP's in A and B respectively such that $D=d+3, d>0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $p-q$ is equal to

If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10},$ then the sum of its first $200$ terms is

The coefficients of three successive terms in the expansion of ${(1 + x)^n}$ are $165, 330$ and $462$ respectively, then the value of n will be

The number of complex numbers z such that $\left| {z - 1\left| = \right|z + 1\left| = \right|z - i} \right|$ equals

The centroid of the triangle formed by joining the feet of the normals drawn from any point to the parabola ${y^2} = 4ax$, lies on

The ${n^{th}}$ term of the following series $(1 \times 3) + (3 \times 5) + (5 \times 7) + (7 \times 9) + .......$ will be

If $\int_a^b {{x^3}dx} = 0$ and $\int_a^b {{x^2}} dx = \frac{2}{3}$, then the value of $a$ and $b$ will be respectively

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

If the tangent to the curve $y^2 = 4x$ at point $(1,2)$ cuts co-ordinate axes at points $A$ and $B$, then area of $\Delta AOB$ is (where $'O'$ is origin)

The value of $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\int_{\frac{\pi }{2}}^x t \,dt}}{{\sin (2x - \pi )}}$ is