How do i solve this question?

∫(cos2xsinx+sinx+1)dx

1 Answer
Øko
Mar 10, 2018

I=−ln(csc(x)+cot(x))+x+C

Explanation:

We want to solve

I=∫cos2(x)sin(x)+sin(x)+1dx

Rewrite the integrand

I=∫1−sin2(x)sin(x)+sin(x)+1dx

I=∫csc(x)−sin(x)+sin(x)+1dx

I=∫csc(x)+1dx

I=∫csc(x)+∫1dx

We know the integral of cosecant as (otherwise look here )

I1=∫csc(x)dx=−ln(|csc(x)+cot(x)|)+C

Thus

I=−ln(|csc(x)+cot(x)|)+x+C

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