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

y^{^{'}} + \frac{4}{x ^{2} - 1} y = 0

\int 4 t^{2} d t

x \to 1 lim (- \frac{1}{x - 1})

\frac{7 + t + t ^{2}}{t}

y = \frac{w ^{4} + 5 w ^{2} + w}{w ^{2}}

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