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

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

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integral of (sin(2θ))/(1+sin^2(θ))

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

ln 1 + sin^{2} (θ) + C

Solution steps

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= ln ∣ u ∣

Substitute back

u = 1 + sin^{2} (θ)

= ln 1 + sin^{2} (θ)

Add a constant to the solution

= ln 1 + sin^{2} (θ) + C

\int 2 + 4 x^{2} d x

\int e^{x} 1 - e^{x} d x

e^{x^{2} y} = x + y

\frac{\partial}{\partial x} (\frac{x}{x ^{2} - y ^{2}})

\frac{d}{d x} (- \frac{8}{1 - 8 x})

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