Solve trigonometric integral \(\displaystyle{\int_{{-\frac{\pi}{{2}}}}^{{\frac{\pi}{{2}}}}}{\frac{{{{\sin}^{{{2014}}}{x}}}}{{{{\sin}^{{{2014}}}{x}}+{{\cos}^{{{2014}}}{x}}}}}{\left.{d}{x}\right.}\)

Kason Palmer

Kason Palmer

Answered question


Solve trigonometric integral

Answer & Explanation



Beginner2022-03-21Added 8 answers

Consider the integral
This may also be seen as
=0π2sin2axsin2ax+cos2ax dxπ20sin2axsin2ax+cos2ax dx
=2,0π2sin2axsin2ax+cos2ax dx
where xx was made in the second integral. Now make the substitution x=tπ2 to obtain
I=2 π20cos2atsin2at+cos2at dt
Now let tx to obtain
I=2 0π2cos2axsin2ax+cos2ax dx
Adding the two integral expressions leads to
=2 0π2sin2ax+cos2axsin2ax+cos2ax dx=2 0π2dx=π.
It can now be started that
π2π2sin2axsin2ax+cos2ax dx=π2

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