What is the integral of-2cot^3(x)sec^2(x) ?

The integral of-2cot^3(x)sec^2(x) is csc^2(x)+C

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integral of-2cot^3(x)sec^2(x)

\int - 2 cot^{3} (x) sec^{2} (x) d x

csc^{2} (x) + C

Solution steps

\int - 2 cot^{3} (x) sec^{2} (x) d x

Take the constant out:

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

= - 2 \cdot \int cot^{3} (x) sec^{2} (x) d x

Rewrite using trig identities

= - 2 \cdot \int cot (x) csc^{2} (x) d x

Apply u-substitution

: \int - \frac{1}{2} d u

= - 2 \cdot \int - \frac{1}{2} d u

Integral of a constant:

\int a d x = a x

= - 2 (- \frac{1}{2}) u

Substitute back

u = csc^{2} (x)

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

Simplify

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

= csc^{2} (x)

Add a constant to the solution

= csc^{2} (x) + C

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What is the integral of-2cot^3(x)sec^2(x) ?
The integral of-2cot^3(x)sec^2(x) is csc^2(x)+C