What is the integral of (1/(sin^2(3x))+3) ?

The integral of (1/(sin^2(3x))+3) is -1/3 cot(3x)+3x+C

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integral of (1/(sin^2(3x))+3)

\int (\frac{1}{sin ^{2} ( 3 x )} + 3) d x

- \frac{1}{3} cot (3 x) + 3 x + C

Solution steps

\int \frac{1}{sin ^{2} ( 3 x )} + 3 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 \frac{1}{sin ^{2} ( 3 x )} d x + \int 3 d x

\int \frac{1}{sin ^{2} ( 3 x )} d x = - \frac{1}{3} cot (3 x)

\int 3 d x = 3 x

= - \frac{1}{3} cot (3 x) + 3 x

Add a constant to the solution

= - \frac{1}{3} cot (3 x) + 3 x + C

\int \frac{sin ( 3 x )}{e ^{2 x}} d x

\int x d tan (x) d x

\int - ln (y) d y

\int (x^{2} + 3)^{3} d x

\int \frac{x - 3}{( x - 1 ) ( x ^{2} - 1 )} d x

What is the integral of (1/(sin^2(3x))+3) ?
The integral of (1/(sin^2(3x))+3) is -1/3 cot(3x)+3x+C