Solving ∫ 2 sin ⁡ ( x ) cos ⁡ ( x ) sin 4...

lilmoore11p8

lilmoore11p8

Answered

2022-07-01

Solving 2 sin ( x ) cos ( x ) sin 4 ( x ) + cos 4 ( x ) d x

Answer & Explanation

Caiden Barrett

Caiden Barrett

Expert

2022-07-02Added 20 answers

Hint
For the denominator
sin 4 ( x ) + cos 4 ( x ) = ( sin 2 ( x ) + cos 2 ( x ) ) 2 2 sin 2 ( x ) cos 2 ( x ) = 1 1 2 sin 2 ( 2 x ) = 1 2 ( 1 + cos 2 ( 2 x ) )
For the numerator
2 sin ( x ) cos ( x ) = sin ( 2 x )
So
2 sin ( x ) cos ( x ) sin 4 ( x ) + cos 4 ( x ) d x = 2 sin ( 2 x ) 1 + cos 2 ( 2 x ) d x
Changing variable t = cos ( 2 x ) looks quite promising.
nidantasnu

nidantasnu

Expert

2022-07-03Added 7 answers

2 sin x cos x sin 4 x + cos 4 x d x = 2 sin x cos x 1 2 sin 2 x cos 2 x d x = sin 2 x 1 1 2 sin 2 2 x d x = sin 2 x 1 + cos 2 2 x d ( 2 x ) = arctan cos 2 x .

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