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

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

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

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

\frac{1}{2} ln 1 + sin^{2} (x) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

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

= \frac{1}{2} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

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

= \frac{1}{2} ln 1 + sin^{2} (x)

Add a constant to the solution

= \frac{1}{2} ln 1 + sin^{2} (x) + C

\frac{d}{d x} (e^{5 x})

y^{^{''}} = y

\int \frac{1}{1 - 6 x ^{2}} d x

\frac{d}{d x} (\frac{x ^{3}}{x + x})

\frac{d}{d x} (x^{3} - 2 x^{2} + x)

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