NIMCET 2020 — Mathematics PYQ
NIMCET | Mathematics | 2020∫f(x),dx=g(x), then ∫x5f(x3),dx
Choose the correct answer:
- A.
31x3g(x3)−3∫x4g(x3),dx+c
31x3g(x3)−∫x2g(x3),dx+c
Explanation
Sol.
(b)
Let x3=t
⇒3x2dx=dt
I=∫x5f(x3),dx
=∫x2x3f(x3),dx
=31∫tf(t),dt
Using integration by parts
I=31[tf(t)dt−∫(dtdtf(t)dt)dt]
=31[tg(t)−∫g(t)dt]
=31tg(t)−31∫g(t)dt
As x3=t and dt=3x2dx
=31x3g(x3)−33∫x2g(x3),dx
=31x3g(x3)−∫x2g(x3),dx+c
Explanation
Sol.
(b)
Let x3=t
⇒3x2dx=dt
I=∫x5f(x3),dx
=∫x2x3f(x3),dx
=31∫tf(t),dt
Using integration by parts
I=31[tf(t)dt−∫(dtdtf(t)dt)dt]
=31[tg(t)−∫g(t)dt]
=31tg(t)−31∫g(t)dt
As x3=t and dt=3x2dx
=31x3g(x3)−33∫x2g(x3),dx
=31x3g(x3)−∫x2g(x3),dx+c

