What is the integral of 1/((tan(x)+cot(x))^2) ?

The integral of 1/((tan(x)+cot(x))^2) is 1/8 (x-1/4 sin(4x))+C

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integral of 1/((tan(x)+cot(x))^2)

integral

\frac{1}{( tan ( x ) + cot ( x ) ) ^{2}}

\frac{1}{8} (x - \frac{1}{4} sin (4 x)) + C

Solution steps

\int \frac{1}{( tan ( x ) + cot ( x ) ) ^{2}} d x

Simplify

\frac{1}{( tan ( x ) + cot ( x ) ) ^{2}} : cos^{2} (x) sin^{2} (x)

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

Rewrite using trig identities

= \int \frac{1 - cos ( 4 x )}{8} d x

Take the constant out:

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

= \frac{1}{8} \cdot \int 1 - cos (4 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{8} (\int 1 d x - \int cos (4 x) d x)

\int 1 d x = x

\int cos (4 x) d x = \frac{1}{4} sin (4 x)

= \frac{1}{8} (x - \frac{1}{4} sin (4 x))

Add a constant to the solution

= \frac{1}{8} (x - \frac{1}{4} sin (4 x)) + C

\frac{1}{x ^{\frac{1}{2}}}

y = x^{2}

\frac{x}{( 1 + x ^{3} )}

\frac{e ^{x}}{( e ^{x} + 1 ) ^{2}}

\frac{x ^{2}}{( x - 1 )}

What is the integral of 1/((tan(x)+cot(x))^2) ?
The integral of 1/((tan(x)+cot(x))^2) is 1/8 (x-1/4 sin(4x))+C