How to integrate ∫ 1 sin 4 ⁡ x cos 4 ⁡ x d x?

veneciasp

veneciasp

Answered

2022-07-10

How to integrate 1 sin 4 x cos 4 x d x?

Answer & Explanation

Elianna Wilkinson

Elianna Wilkinson

Expert

2022-07-11Added 11 answers

By using the substitution x = arctan t we have d x = d t 1 + t 2 and sin 2 ( x ) = t 2 1 + t 2 , cos 2 ( x ) = 1 1 + t 2 , hence
d x sin 4 ( x ) cos 4 ( x ) = ( 1 + t 2 ) 3 t 4 d t
and the last integral is straightforward to compute through the binomial formula. It equals:
C 1 3 t 3 3 t + 3 t + t 3 3 .

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