What is the integral of (cos(x)sin(x))/(1+cos^4(x)) ?

The integral of (cos(x)sin(x))/(1+cos^4(x)) is -1/2 arctan(cos^2(x))+C

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integral of (cos(x)sin(x))/(1+cos^4(x))

\int \frac{cos ( x ) sin ( x )}{1 + cos ^{4} ( x )} d x

- \frac{1}{2} arctan (cos^{2} (x)) + C

Solution steps

\int \frac{cos ( x ) sin ( x )}{1 + cos ^{4} ( x )} d x

Apply u-substitution

: \int - \frac{u}{1 + u ^{4}} d u

= \int - \frac{u}{1 + u ^{4}} d u

Take the constant out:

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

= - \int \frac{u}{1 + u ^{4}} d u

Apply u-substitution

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

= - \int \frac{1}{2 ( 1 + v ^{2} )} d v

Take the constant out:

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

= - \frac{1}{2} \cdot \int \frac{1}{1 + v ^{2}} d v

Use the common integral:

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

= - \frac{1}{2} arctan (v)

Substitute back

= - \frac{1}{2} arctan (cos^{2} (x))

Add a constant to the solution

= - \frac{1}{2} arctan (cos^{2} (x)) + C

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What is the integral of (cos(x)sin(x))/(1+cos^4(x)) ?
The integral of (cos(x)sin(x))/(1+cos^4(x)) is -1/2 arctan(cos^2(x))+C