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

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

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integral of (cos(2x))/(cos^2(x)sin^2(x))

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

- cot (x) - tan (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{cos ^{2} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x ) sin ^{2} ( x )} : \frac{1}{sin ^{2} ( x )} - \frac{1}{cos ^{2} ( x )}

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

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

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

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

= - cot (x) - tan (x)

Add a constant to the solution

= - cot (x) - tan (x) + C

\int \frac{1}{( u ^{2} + a ^{2} ) ^{2}} d u

n = 3 \sum \infty (\frac{- 1}{2})^{n}

y^{^{''}} + 10 y^{^{'}} + 26 y = 0

f (x) = x^{2} 9 - x^{2}

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

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