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

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

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

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

- csc (x) + C

Solution steps

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

Simplify

\frac{csc ^{2} ( x )}{sec ( x )} : cot (x) csc (x)

= \int cot (x) csc (x) d x

Apply u-substitution

: \int - 1 d u

= \int - 1 d u

Integral of a constant:

\int a d x = a x

= (- 1) u

Substitute back

u = csc (x)

= (- 1) csc (x)

Simplify

= - csc (x)

Add a constant to the solution

= - csc (x) + C

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