Question

Evaluate the integral

Answer 1

e^(sin x) + (x^(1 + e))/(1 + e) + c

Explanation:

split up the integral:

∫(e^sinx)/sec x dx + ∫x^e dx.

let's focus on ∫(e^sinx)/sec x dx first.

∫(e^sinx)/sec x dx = ∫cos x (e^sinx) dx.

use a sub.

let u = sin x.

du/dx = cos x, dx = du/cos x

now we have ∫cos x (e^u) du/cos x

= ∫(cos x (e^u) du) / cos x

= ∫(e^u) du

= e^u

= e^sin x

that is answer to first integral.

now for second integral:

∫x^e dx.

just pretend like the e is a regular number.

∫x^e dx

= (1/(1 + e)) x^(1 + e)

= (x^(1 + e)) / (1 + e)

that is second integral.

so, the answer to the question is e^sin x + (x^(1 + e)) /(1+e) + c