Integral &#x222B;<!-- ∫ --> d x </mrow> sin

Makayla Boyd 2022-06-25 Answered
Integral d x sin 2 ( x ) + sin ( 2 x )
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Answers (1)

kejohananws
Answered 2022-06-26 Author has 19 answers
d x sin 2 ( x ) + sin ( 2 x ) = sec 2 ( x ) sin 2 ( x ) sec 2 ( x ) + 2 sin ( x ) cos ( x ) sec 2 ( x ) d x = sec 2 ( x ) tan 2 ( x ) + 2 tan ( x ) d x
Letting u := tan ( x ), then d u = sec 2 ( x ) d x gives
d x sin 2 ( x ) + sin ( 2 x ) = d u u 2 + 2 u = d u ( u + 1 ) 2 1
Letting z:=u+1, then dz=du gives
d x sin 2 ( x ) + sin ( 2 x ) = d z z 2 1 = artanh  ( z ) + C = artanh  ( u + 1 ) + C
Reersing the final substitution gives
d x sin 2 ( x ) + sin ( 2 x ) = artanh  ( tan ( x ) + 1 ) + C

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