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

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

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

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

\frac{1}{2} arctan (sin^{2} (z)) + C

Solution steps

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

Apply u-substitution

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

= \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 (sin^{2} (z))

Add a constant to the solution

= \frac{1}{2} arctan (sin^{2} (z)) + C

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