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

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

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

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

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

Solution steps

\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

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

x \to 1 + lim (x^{2} - 4 x + 3)

\frac{\partial}{\partial x} (4 e^{\frac{x ^{2}}{y}} - cos (x y^{3}))

x \to 0 lim (4 x + x)

x, y = x^{2}, x = y^{2}

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