What is ∫sin³xcosx dx equal to?

Correct Answer is : (1−cos2x)24+c{(1-cos²x)²}{4}+c4(1−cos2x)2+c

Correct Answer is : (1−cos2x)24+c{(1-cos²x)²}{4}+c4(1−cos2x)2+c

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UPSC NDA II 2018 Mathematics Solved Paper

Question : 113 of 120

Marks: +1, -0

What is

∫sin^{3}xcosx dx

equal to?

Solution:

Concept:

Integration by substitution using the power rule

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Explanation:

Let

I = \int \sin^3 x \cos x \, dx

Substitute

t = \sin x

Then

dt = \cos x \, dx

The integral becomes

\int t^3 \, dt = \frac{t^4}{4} + C

Substitute back:

I = \frac{\sin^4 x}{4} + C

Using

\sin^2 x = 1 - \cos^2 x

, we get

I = \frac{(1 - \cos^2 x)^2}{4} + C

This matches option D.

Answer:

Option D:

\frac{(1 - \cos^2 x)^2}{4} + C

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