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

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

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

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

- \frac{1}{2} ln ∣ cos (2 x) ∣ + C

Solution steps

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

Rewrite using trig identities

= \int tan (2 x) d x

Rewrite using trig identities

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

Apply u-substitution

: \int - \frac{1}{2 u} d u

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

Take the constant out:

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

= - \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 = cos (2 x)

= - \frac{1}{2} ln ∣ cos (2 x) ∣

Add a constant to the solution

= - \frac{1}{2} ln ∣ cos (2 x) ∣ + C

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