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

The integral of (1+1/(sin^2(x)))cos(x) is sin(x)-csc(x)+C

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

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

sin (x) - csc (x) + C

Solution steps

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

Expand

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

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

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

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

= sin (x) - csc (x)

Add a constant to the solution

= sin (x) - csc (x) + C

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