Correct Answer is : x2sin⁡x+2xcos⁡x−2sin⁡x+cx² x+2 x x-2 x+cx2sinx+2xcosx−2sinx+c, where c is the constant of integration

x² x+2 x x-2 x+c, where c is the constant of integration

x² x-2 x x-2 x+c, where c is the constant of integration

x² x-2 x x+2 x+c, where c is the constant of integration

x² x+2 x x+2 x+c, where c is the constant of integration

Correct Answer is : x2sin⁡x+2xcos⁡x−2sin⁡x+cx² x+2 x x-2 x+cx2sinx+2xcosx−2sinx+c, where c is the constant of integration

Test Index

MHTCET 26 Apr 2025 Shift 2 Paper

Question : 132 of 150

Marks: +1, -0

\int x^{2} \cos x d x=

Solution:

Concept:

Use integration by parts, choosing

x^2

as the first function and

\cos x

as the second function.

Explanation:

Let

I = \int x^2 \cos x \, dx

Applying integration by parts:

I = x^2 \sin x - \int 2x \sin x \, dx

I = x^2 \sin x - 2\int x \sin x \, dx

Now integrate

\int x \sin x \, dx

again by parts:

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

= -x \cos x + \sin x

Substituting back:

I = x^2 \sin x - 2(-x \cos x + \sin x) + c

I = x^2 \sin x + 2x \cos x - 2\sin x + c

Answer:

x^2 \sin x + 2x \cos x - 2\sin x + c

Hence, the correct option is A.

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