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

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

\int_{0}^{4} \frac{x}{2} d x

(1 - x^{2}) y^{^{'}} - x y = 1

\frac{2 x}{x ^{2} - 7}

y^{^{'}} + 6 y = 3

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