If f(x) d x=g(x) then x³ f≤ft(x²) d x is equal to

Correct Answer is : 12[x2g(x2)−∫g(x2)d(x2)]1/2≤ft[x² g≤ft(x²)- g≤ft(x²) d≤ft(x²)]21[x2g(x2)−∫g(x2)d(x2)]

1/2≤ft[x² g≤ft(x²)- g≤ft(x²) d≤ft(x²)]

1/2≤ft[x²[f(x)]²-[g(x)]² d x]

1/2≤ft[x² g≤ft(x²)+ g≤ft(x²) d≤ft(x²)]

Correct Answer is : 12[x2g(x2)−∫g(x2)d(x2)]1/2≤ft[x² g≤ft(x²)- g≤ft(x²) d≤ft(x²)]21[x2g(x2)−∫g(x2)d(x2)]

Test Index

MHTCET 17 Apr 2026 Shift 2 Paper

Question : 128 of 150

Marks: +1, -0

\int f(x) d x=g(x)

\int x^{3} f\left(x^{2}\right) d x

is equal to

$\frac{1}{2}\left[x^{2} g\left(x^{2}\right)-\int g\left(x^{2}\right) d\left(x^{2}\right)\right]$
$\frac{1}{2}\left[x^{2}[f(x)]^{2}-\int[g(x)]^{2} d x\right]$
$\frac{1}{2}\left[x^{2} g(x)-\int g(x) d(x)\right]$
$\frac{1}{2}\left[x^{2} g\left(x^{2}\right)+\int g\left(x^{2}\right) d\left(x^{2}\right)\right]$

Solution:

Concept:

Use substitution followed by integration by parts to simplify the integral.

Explanation:

Given

\int f(x)\,dx = g(x)

, we have

g'(x)=f(x)

Let

t=x^2

Then

dt=2x\,dx

, so

x^3\,dx = x^2 \cdot x\,dx = \frac{t}{2}\,dt

The integral becomes

\frac{1}{2}\int t\,f(t)\,dt

Using integration by parts, taking

u=t

and

dv=f(t)\,dt

\int t\,f(t)\,dt = t\,g(t) - \int g(t)\,dt

Substitute

t=x^2

back, noting

dt=d(x^2)

Therefore,

\int x^3 f(x^2)\,dx = \frac{1}{2}\left[x^2 g(x^2) - \int g(x^2)\,d(x^2)\right]

This matches option A.

Answer:

Option A.

Go to Question:

Prev Question

Next Question