NIMCET 2020 — Mathematics PYQ

Sol.
(b)
Let $x^3=t$
$\Rightarrow 3x^{2}dx=dt$
$I=\int x^{5}f(x^3),dx$
$=\int x^2{x}^{3}f(x^3),dx$
$=\frac{1}{3}\int tf(t),dt$

Using integration by parts

$I=\frac{1}{3}\left[tf(t)dt-\int \left(\frac{dt}{dt}f(t)dt\right)dt\right]$

$=\frac{1}{3}[tg(t)-\int g(t)dt]$

$=\frac{1}{3}tg(t)-\frac{1}{3}\int g(t)dt$

As $x^3=t$ and $dt=3x^{2}dx$

$=\frac{1}{3}{x}^{3}g(x^3)-\frac{3}{3}\int x^{2}g(x^3),dx$

$=\frac{1}{3}{x}^{3}g(x^3)-\int x^{2}g(x^3),dx+c$