What is the integral of (sin(2x))/(12+cos^2(x)) ?

The integral of (sin(2x))/(12+cos^2(x)) is -ln|cos(2x)+25|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sin(2x))/(12+cos^2(x))

\int \frac{sin ( 2 x )}{12 + cos ^{2} ( x )} d x

- ln ∣ cos (2 x) + 25 ∣ + C

Solution steps

\int \frac{sin ( 2 x )}{12 + cos ^{2} ( x )} d x

Rewrite using trig identities

= \int \frac{2 sin ( 2 x )}{cos ( 2 x ) + 25} 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 ) + 25} 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) + 25

= 2 (- \frac{1}{2} ln ∣ cos (2 x) + 25 ∣)

Simplify

2 (- \frac{1}{2} ln ∣ cos (2 x) + 25 ∣) : - ln ∣ cos (2 x) + 25 ∣

= - ln ∣ cos (2 x) + 25 ∣

Add a constant to the solution

= - ln ∣ cos (2 x) + 25 ∣ + C

\frac{2}{( s ^{3} - 7 s ^{2} + 2 s + 40 )}

arccos (∣ x + 1 ∣)

f (x) = 2 x^{2} - 5 x + 3, at a = 2

\frac{\partial}{\partial x} (ln (2 x^{2} + x y))

\int \frac{1}{x 100 x ^{2} - 100} d x

What is the integral of (sin(2x))/(12+cos^2(x)) ?
The integral of (sin(2x))/(12+cos^2(x)) is -ln|cos(2x)+25|+C