Help with \(\displaystyle\int{\frac{{{1}+{\sin{{x}}}}}{{{\left({1}+{2}{\cos{{x}}}\right)}{\sin{{x}}}}}}{\left.{d}{x}\right.}\)

Tony Mccarthy

Tony Mccarthy

Answered question

2022-03-24

Help with 1+sinx(1+2cosx)sinxdx

Answer & Explanation

Cason Singleton

Cason Singleton

Beginner2022-03-25Added 13 answers

Trigonometric functions can be expressed using rational functions t=tanx2.Here, specifically, we must
sinx=2t1+t2, cosx=1t21+t2
So we proceed by substitution: if
 dt =12(1+tan2x2)dxdx=2 dt 1+tan2x and
1+sinx(1+2cosx)sinx dx =1+2t1+t2(1+2(1t2)1+t2)2t1+t22 dt 1+t2
=(1+t2+2t)2 dt (1+t2+2(1t2))t=2(1+t)2t(3t2) dt 
The integral of a rational function is now in front of you, and it needs to be divided into partial fractions as follows:
2(1+t)2t(3t2)=At+Bt3t2 (A,BR)

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