What is the integral of (cos(x)sin(x))/(1+sin^2(x)) ?

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

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integral of (cos(x)sin(x))/(1+sin^2(x))

\int \frac{cos ( x ) sin ( x )}{1 + sin ^{2} ( x )} d x

\frac{1}{2} ln 1 + sin^{2} (x) + C

Solution steps

\int \frac{cos ( x ) sin ( x )}{1 + sin ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{sin ( 2 x )}{2 ( 1 + sin ^{2} ( x ) )} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int \frac{sin ( 2 x )}{1 + sin ^{2} ( x )} d x

Apply u-substitution

: \int \frac{1}{u} d u

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

u = 1 + sin^{2} (x)

= \frac{1}{2} ln 1 + sin^{2} (x)

Add a constant to the solution

= \frac{1}{2} ln 1 + sin^{2} (x) + C

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