The value of integral x³ x d x is...

Correct Answer is : x3sin⁡x+3x2cos⁡x−6xsin⁡x−6cos⁡x+cx³ x+3 x² x-6 x x-6 x+cx3sinx+3x2cosx−6xsinx−6cosx+c

Correct Answer is : x3sin⁡x+3x2cos⁡x−6xsin⁡x−6cos⁡x+cx³ x+3 x² x-6 x x-6 x+cx3sinx+3x2cosx−6xsinx−6cosx+c

Test Index

MHTCET 19 Apr 2026 Shift 2 Paper

Question : 127 of 150

Marks: +1, -0

\int x^{3} \cos x d x

is...

Solution:

Concept:

Integration by parts is applied repeatedly to reduce the power of

x

until only a simple integral remains.

Explanation:

Use the formula

\int u\,dv = uv - \int v\,du

Take

u = x^{3}

and

dv = \cos x\,dx

Then

du = 3x^{2}\,dx

and

v = \sin x

So,

\int x^{3} \cos x\,dx = x^{3} \sin x - 3\int x^{2} \sin x\,dx

Apply parts again on

\int x^{2} \sin x\,dx

with

u = x^{2}

and

dv = \sin x\,dx

This gives

-x^{2} \cos x + 2\int x \cos x\,dx

Now

\int x \cos x\,dx = x \sin x + \cos x + c

Substituting back, the full integral simplifies to

x^{3} \sin x + 3x^{2} \cos x - 6x \sin x - 6 \cos x + c

This matches option C.

Answer:

Option C

Go to Question:

Prev Question

Next Question