What is the integral of csc(x)(2csc(x)+3cot(x)) ?

The integral of csc(x)(2csc(x)+3cot(x)) is -cot(x)(2+3cos(x))-3sin(x)+C

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integral of csc(x)(2csc(x)+3cot(x))

\int csc (x) (2 csc (x) + 3 cot (x)) d x

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

Solution steps

\int csc (x) (2 csc (x) + 3 cot (x)) d x

Rewrite using trig identities

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

Apply Integration By Parts

= - cot (x) (2 + 3 cos (x)) - \int 3 sin (x) cot (x) d x

\int 3 sin (x) cot (x) d x = 3 sin (x)

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

Add a constant to the solution

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

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\int \frac{ln ( x + 1 ) - ln ( x )}{x ( x + 1 )} d x

\int (x^{2} + e^{x}) d x

\int - 9 sin^{2} (x) d x

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