What is the integral of (2sin^3(x)-3)/(1-cos^2(x)) ?

The integral of (2sin^3(x)-3)/(1-cos^2(x)) is -2cos(x)+3cot(x)+C

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integral of (2sin^3(x)-3)/(1-cos^2(x))

\int \frac{2 sin ^{3} ( x ) - 3}{1 - cos ^{2} ( x )} d x

- 2 cos (x) + 3 cot (x) + C

Solution steps

\int \frac{2 sin ^{3} ( x ) - 3}{1 - cos ^{2} ( x )} d x

Expand

\frac{2 sin ^{3} ( x ) - 3}{1 - cos ^{2} ( x )} : \frac{2 sin ^{3} ( x )}{1 - cos ^{2} ( x )} - \frac{3}{1 - cos ^{2} ( 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{2 sin ^{3} ( x )}{1 - cos ^{2} ( x )} d x - \int \frac{3}{1 - cos ^{2} ( x )} d x

\int \frac{2 sin ^{3} ( x )}{1 - cos ^{2} ( x )} d x = - 2 cos (x)

\int \frac{3}{1 - cos ^{2} ( x )} d x = - 3 cot (x)

= - 2 cos (x) - (- 3 cot (x))

Simplify

= - 2 cos (x) + 3 cot (x)

Add a constant to the solution

= - 2 cos (x) + 3 cot (x) + C

\int \frac{x - x cos ( a )}{x} d x

\int \frac{1}{x ( x ^{2} + 3 )} d x

\int \frac{x ^{3} + 4 x ^{2} + 8 x + 5}{x ^{2} + 2 x + 1} d x

\int \frac{1}{4 x ^{2} + 4 x + 2} d x

\int \frac{tan ^{5} ( ln ( x ) )}{x} d x

What is the integral of (2sin^3(x)-3)/(1-cos^2(x)) ?
The integral of (2sin^3(x)-3)/(1-cos^2(x)) is -2cos(x)+3cot(x)+C