Question

integral e power of sinx

Answer 1

Here it is:

We open e^x to Maclaurin series and integrate term by term. It is a long way though.

e^x = 1 + x + x^2/2 + x^3/3! + ... + x^n/n! ...

f(x) = e^sin(x) = 1 + sin(x) + sin^2(x)/2 + sin^3(x)/6 + sin^4(x)/24 + ...

int f(x) dx = x - cos(x) - sin(2x)/4 + x/2 - sin^2(x)cos(x)/3 - 2cos(x)/3 ... + C

By the way

int sin^m(x) dx = -sin^(m-1)(x) cos(x)/m + (m-1)int(sin^(m-2))dx /m