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

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

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

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

\frac{1}{2} ln 1 + sin^{2} (θ) + C

Solution steps

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

Rewrite using trig identities

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

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 θ )}{1 + sin ^{2} ( θ )} d θ

Apply u-substitution

= \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} (θ)

= \frac{1}{2} ln 1 + sin^{2} (θ)

Add a constant to the solution

= \frac{1}{2} ln 1 + sin^{2} (θ) + C

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