Q51 / 90 · JEE Main 2019 (08 Apr Shift 2) · Mathematics

If $\int \frac{d x}{x^{3} {(1 + x^{6})}^{\frac{2}{3}}} = x f (x) {(1 + x^{6})}^{\frac{1}{3}} + C$ , where $C$ is a constant of integration, then the function $f (x)$ is equal to

A
$\frac{3}{x^{2}}$
Correct answer
B
$- \frac{1}{2 x^{3}}$
C
$- \frac{1}{6 x^{3}}$
D
$- \frac{1}{2 x^{2}}$

Solution

We have, $\int \frac{d x}{x^{3} {(1 + x^{6})}^{\frac{2}{3}}} = x f (x) {(1 + x^{6})}^{\frac{1}{3}} + C$

Now, $I = \int \frac{d x}{x^{7} {(\frac{1}{x^{6}} + 1)}^{\frac{2}{3}}}$

Put, $t = \frac{1}{x^{6}} + 1$ $\Rightarrow d t = - \frac{6}{x^{7}} d x$

$∴ I = - \frac{1}{6} \int \frac{d t}{t^{\frac{2}{3}}} = - \frac{1}{2} t^{\frac{1}{3}} + C$ , where $C$ is the constant of integration $= - \frac{1}{2} {(\frac{1}{x^{6}} + 1)}^{\frac{1}{3}} + C = - \frac{1}{2} \frac{{(1 + x^{6})}^{\frac{1}{3}}}{x^{2}} + C$

$∴ f (x) = - \frac{1}{2 x^{3}}$

Hence, the correct answer is option: 'B'.

If ∫dxx31+x623 =xfx1+x613+C, where C is a constant of integration, then the function fx is equal to

Solution

Relevant Questions

If $\int \frac{d x}{x^{3} {(1 + x^{6})}^{\frac{2}{3}}} = x f (x) {(1 + x^{6})}^{\frac{1}{3}} + C$ , where $C$ is a constant of integration, then the function $f (x)$ is equal to