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

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

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

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

sec (x) + C

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply the Power Rule

: - \frac{1}{u}

= - (- \frac{1}{u})

Substitute back

u = cos (x)

= - (- \frac{1}{cos ( x )})

Simplify

- (- \frac{1}{cos ( x )}) : sec (x)

= sec (x)

Add a constant to the solution

= sec (x) + C

\int sech^{2} (2 x) d x

\int \frac{x ^{3}}{3 - x ^{2}} d x

\int (x - 1)^{- 1} d x

\int \frac{- 2 x + 6}{( x + 1 ) ( x ^{2} + x + 1 )} d x

\int \frac{x ^{2}}{( x ^{2} + 9 ) ^{\frac{3}{2}}} d x

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