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

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

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integral of csc(x)(sin(x)+cot(x))

\int csc (x) (sin (x) + cot (x)) d x

x - csc (x) + C

Solution steps

\int csc (x) (sin (x) + cot (x)) d x

Rewrite using trig identities

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

Expand

\frac{sin ^{2} ( x ) + cos ( x )}{sin ^{2} ( x )} : 1 + \frac{cos ( x )}{sin ^{2} ( x )}

= \int 1 + \frac{cos ( x )}{sin ^{2} ( 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 1 d x + \int \frac{cos ( x )}{sin ^{2} ( x )} d x

\int 1 d x = x

\int \frac{cos ( x )}{sin ^{2} ( x )} d x = - csc (x)

= x - csc (x)

Add a constant to the solution

= x - csc (x) + C

\int \frac{x - 1}{x ( x - 2 )} d x

\int sin^{2} (\frac{3 π x}{L}) d x

\int \frac{ln ( 2 x )}{x} d x

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

\int \frac{( ln ( x ) ) ^{2}}{x} d x

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