Question

Integrate \(3x^2 (cos⁡x^3+8)\).

(a) \(sin⁡x^3-8x^3+C\)

(b) \(sin⁡x^3+8x^3+C\)

(c) –\(sin⁡x^3+8x^3+C\)

(d) \(sin⁡x^3-x^3+C\)

This question was addressed to me during an online interview.

I'm obligated to ask this question of Methods of Integration-1 topic in section Integrals of Mathematics – Class 12

Answer 1

The correct answer is (b) \(sin⁡x^3+8x^3+C\)

Easy explanation: By using the method of integration by substitution,

Let x^3=t

Differentiating w.r.t x, we get

3x^2 dx=dt

\(\int 3x^2 \,(cos⁡x^3+8) \,dx=\int (cos⁡t+8)dt\)

\(\int (cos⁡t+8) dt=sin⁡t+8t\)

Replacing t with x^3,we get

\(\int 3x^2 (cos⁡x^3+8) dx=sin⁡x^3+8x^3+C\)