What is the integral of-csc(5x+6)cot(5x+6) ?

The integral of-csc(5x+6)cot(5x+6) is 1/5 cos(5x+6)cot(5x+6)+1/5 sin(5x+6)+C

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integral of-csc(5x+6)cot(5x+6)

\int - csc (5 x + 6) cot (5 x + 6) d x

\frac{1}{5} cos (5 x + 6) cot (5 x + 6) + \frac{1}{5} sin (5 x + 6) + C

Solution steps

\int - csc (5 x + 6) cot (5 x + 6) d x

Take the constant out:

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

= - \int csc (5 x + 6) cot (5 x + 6) d x

Rewrite using trig identities

= - \int \frac{cos ( 5 x + 6 )}{sin ^{2} ( 5 x + 6 )} d x

Apply Integration By Parts

: - \frac{1}{5} cos (5 x + 6) cot (5 x + 6) - \int sin (5 x + 6) cot (5 x + 6) d x

= - (- \frac{1}{5} cos (5 x + 6) cot (5 x + 6) - \int sin (5 x + 6) cot (5 x + 6) d x)

\int sin (5 x + 6) cot (5 x + 6) d x = \frac{1}{5} sin (5 x + 6)

= - (- \frac{1}{5} cos (5 x + 6) cot (5 x + 6) - \frac{1}{5} sin (5 x + 6))

Simplify

= \frac{1}{5} cos (5 x + 6) cot (5 x + 6) + \frac{1}{5} sin (5 x + 6)

Add a constant to the solution

= \frac{1}{5} cos (5 x + 6) cot (5 x + 6) + \frac{1}{5} sin (5 x + 6) + C

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What is the integral of-csc(5x+6)cot(5x+6) ?
The integral of-csc(5x+6)cot(5x+6) is 1/5 cos(5x+6)cot(5x+6)+1/5 sin(5x+6)+C