Solve this integral: \int_0^{\pi}\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}xdx

Arslan Lyons

Arslan Lyons

Answered question

2022-02-27

Solve this integral:
0π3cosx+8+cos2xsinxxdx

Answer & Explanation

mljaskaj0u

mljaskaj0u

Beginner2022-02-28Added 2 answers

Here's one way to go.
First, note that
0π3cosx+8+cos2xsinxxdx=0π3x(1+cosx)sinxdx+0π3xsinx(1+1sin2x9)dx
For now I'll simply claim that
0π3x(1+cosx)sinxdx=πlog64
But
0π(1+1sin2x9)dx=0π3xsinxk=1(12k)(1)k32ksin2kx
=k=1(12k)(1)k32k10πxsin2k1xdx
=k=1(12k)(1)k32k1π32Γ(k)2Γ(k+12)
=πk=1132k12k(2k1)
=πlog3227

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