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

The integral of (1/(sin^2(x))+1)sin(x)) is d(ln|tan(x/2)|-cos(x))+C

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

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

d (ln tan (\frac{x}{2}) - cos (x)) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Expand

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

= d \cdot \int \frac{1}{sin ( x )} + sin (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

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

\int \frac{1}{sin ( x )} d x = ln tan (\frac{x}{2})

\int sin (x) d x = - cos (x)

= d (ln tan (\frac{x}{2}) - cos (x))

Add a constant to the solution

= d (ln tan (\frac{x}{2}) - cos (x)) + C

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

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\int 6 x x d x

What is the integral of (1/(sin^2(x))+1)sin(x)) ?
The integral of (1/(sin^2(x))+1)sin(x)) is d(ln|tan(x/2)|-cos(x))+C