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

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

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

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

- ln 1 + cos^{2} (x) + C

Solution steps

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

Simplify

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

Apply u-substitution

: \int - \frac{1}{u} d u

= \int - \frac{1}{u} d u

Take the constant out:

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

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

u = 1 + cos^{2} (x)

= - ln 1 + cos^{2} (x)

Add a constant to the solution

= - ln 1 + cos^{2} (x) + C

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\int x^{3} \frac{1}{x} d x

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