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Question
\[\int \cos^5 x \text{ dx }\]
Sum
Solution
∫ cos5 x dx
= ∫ cos4 x . cos x dx
= ∫ (1 – sin2 x)2 cos x dx
Let sin x = t
⇒ cos x dx = dt
Now, ∫ (1 – sin2 x)2 cos x dx
= ∫ (1 – t2)2 . dt
= ∫ (1 + t4 – 2t2) dt
= ∫ dt + ∫ t4 dt – 2 ∫t2 dt
\[= t + \frac{t^5}{5} - \frac{2 t^3}{3} + C\]
\[ = \sin x + \frac{\sin^5 x}{5} - \frac{2}{3} \sin^3 x + C\]
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