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Question
\[\int \sin^3 x \cos^5 x \text{ dx }\]
Sum
Solution
∫ sin3 x . cos5 x dx
= ∫ sin2 x . cos5 x . sin x dx
= ∫ (1 – cos2 x) . cos5 x sin x dx
Let cos x = t
⇒ – sin x dx = dt
⇒ sin x dx = – dt
Now, ∫ (1 – cos2 x) . cos5 x sin x dx
= –∫ (1 – t2) t5 dt
= –∫ (t5 – t7) dt
= ∫(t7 – t5) dt
\[= \frac{t^8}{8} - \frac{t^6}{6} + C\]
\[ = \frac{\cos^8 x}{8} - \frac{\cos^6 x}{6} + C\]
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