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

The integral of (3cos(x))/(1+sin^2(x)) is 3arctan(sin(x))+C

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

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

3 arctan (sin (x)) + C

Solution steps

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

Take the constant out:

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

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

Apply u-substitution

= 3 \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 3 arctan (u)

Substitute back

u = sin (x)

= 3 arctan (sin (x))

Add a constant to the solution

= 3 arctan (sin (x)) + C

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