How do i solve this question?

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

1 Answer
Øko
Mar 10, 2018

I=-ln(csc(x)+cot(x))+x+C

Explanation:

We want to solve

I=intcos^2(x)/sin(x)+sin(x)+1dx

Rewrite the integrand

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

color(white)(I)=intcsc(x)-sin(x)+sin(x)+1dx

color(white)(I)=intcsc(x)+1dx

color(white)(I)=intcsc(x)+int1dx

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

I_1=intcsc(x)dx=-ln(abs(csc(x)+cot(x)))+C

Thus

I=-ln(abs(csc(x)+cot(x)))+x+C

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