What is the integral of (sin(2x))/(5+cos^2(x)) ?

The integral of (sin(2x))/(5+cos^2(x)) is -ln|cos(2x)+11|+C

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integral of (sin(2x))/(5+cos^2(x))

\int \frac{sin ( 2 x )}{5 + cos ^{2} ( x )} d x

- ln ∣ cos (2 x) + 11 ∣ + C

Solution steps

\int \frac{sin ( 2 x )}{5 + cos ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{2 sin ( 2 x )}{cos ( 2 x ) + 11} d x

Take the constant out:

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

= 2 \cdot \int \frac{sin ( 2 x )}{cos ( 2 x ) + 11} d x

Apply u-substitution

: \int - \frac{1}{2 u} d u

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

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 (- \frac{1}{2} ln ∣ u ∣)

Substitute back

u = cos (2 x) + 11

= 2 (- \frac{1}{2} ln ∣ cos (2 x) + 11 ∣)

Simplify

2 (- \frac{1}{2} ln ∣ cos (2 x) + 11 ∣) : - ln ∣ cos (2 x) + 11 ∣

= - ln ∣ cos (2 x) + 11 ∣

Add a constant to the solution

= - ln ∣ cos (2 x) + 11 ∣ + C

x \to - \infty lim (- x^{3})

\frac{\partial}{\partial y} (x^{a} y^{1 - a})

\frac{d}{d y} (y + 2 + y \frac{y + 2}{2} - \frac{( y + 2 ) ^{2}}{4})

x \to - 4 lim (x^{2} - 16)

\frac{d}{d x} (2 e)

What is the integral of (sin(2x))/(5+cos^2(x)) ?
The integral of (sin(2x))/(5+cos^2(x)) is -ln|cos(2x)+11|+C