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

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

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integral of (sin(2x))/(1-sin^2(x))

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

- ln (cos^{2} (x)) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Take the constant out:

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = cos^{2} (x)

= - ln cos^{2} (x)

Apply absolute rule:

x^{2} = x^{2}

, assuming

x \in R

cos^{2} (x) = cos^{2} (x)

= - ln (cos^{2} (x))

Add a constant to the solution

= - ln (cos^{2} (x)) + C

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