What is the integral of (cos(u))/(sin(u)+sin^3(u)) ?

The integral of (cos(u))/(sin(u)+sin^3(u)) is ln|sin(u)|-1/2 ln|sin^2(u)+1|+C

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integral of (cos(u))/(sin(u)+sin^3(u))

\int \frac{cos ( u )}{sin ( u ) + sin ^{3} ( u )} d u

ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1 + C

Solution steps

\int \frac{cos ( u )}{sin ( u ) + sin ^{3} ( u )} d u

Apply u-substitution

: \int \frac{1}{v + v ^{3}} d v

= \int \frac{1}{v + v ^{3}} d v

Take the partial fraction of

\frac{1}{v + v ^{3}} : \frac{1}{v} - \frac{v}{v ^{2} + 1}

= \int \frac{1}{v} - \frac{v}{v ^{2} + 1} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{v} d v - \int \frac{v}{v ^{2} + 1} d v

\int \frac{1}{v} d v = ln ∣ v ∣

\int \frac{v}{v ^{2} + 1} d v = \frac{1}{2} ln v^{2} + 1

= ln ∣ v ∣ - \frac{1}{2} ln v^{2} + 1

Substitute back

v = sin (u)

= ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1

Add a constant to the solution

= ln ∣ sin (u) ∣ - \frac{1}{2} ln sin^{2} (u) + 1 + C

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What is the integral of (cos(u))/(sin(u)+sin^3(u)) ?
The integral of (cos(u))/(sin(u)+sin^3(u)) is ln|sin(u)|-1/2 ln|sin^2(u)+1|+C