What is the integral of (sec^2(x))/(csc(x)) ?

The integral of (sec^2(x))/(csc(x)) is sec(x)+C

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integral of (sec^2(x))/(csc(x))

\int \frac{sec ^{2} ( x )}{csc ( x )} d x

sec (x) + C

Solution steps

\int \frac{sec ^{2} ( x )}{csc ( x )} d x

Rewrite using trig identities

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

Expand

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

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

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

\int \frac{tan ^{2} ( x )}{csc ( x )} d x = cos (x) + sec (x)

= - cos (x) + cos (x) + sec (x)

Simplify

= sec (x)

Add a constant to the solution

= sec (x) + C

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\int_{- 3}^{2} (6 - x - x^{2}) d x

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