Compute \int \sin(x) ( \frac{1}{\cos(x) + \sin(x)} + \frac{1}{\cos(x) -

wingtipti

wingtipti

Answered question

2022-01-29

Compute sin(x)(1cos(x)+sin(x)+1cos(x)sin(x))dx

Answer & Explanation

Darian Sosa

Darian Sosa

Beginner2022-01-30Added 12 answers

Multiplying the fractions is actually the way to go and everything cancels out!
sinx(2cosxcos2xsin2x)dx=sin2xcos2xdx=tan2xdx
Now, you just need to find the antiderivative of tan
By the way: partial fraction decomposition is only done for functions with polynomials as numerators and denominators. Unless you can make a genius substitution which transforms the integral to a rational function, partial fractions is irrelevant. Usually, when confronted with completely trigonometric integrals, the first thing you should try is to simplify the integrand using trigonometric identities; only then do you want to consider substitutions or integration by parts.

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