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

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

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

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

- 2 cot (x) + 3 x + C

Solution steps

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

Expand

\frac{2 + 3 sin ^{2} ( x )}{sin ^{2} ( x )} : \frac{2}{sin ^{2} ( x )} + 3

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

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

\int 3 d x = 3 x

= - 2 cot (x) + 3 x

Add a constant to the solution

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

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

\int x \frac{1}{x ^{2}} d x

\frac{d}{d x} (\frac{4 x}{x ^{2} - 2})

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

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

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