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

The integral of (1+cos^2(x))/(1+cos(2x)) is 1/2 (tan(x)+x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{2} (tan (x) + x) + C

Solution steps

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

Rewrite using trig identities

= \int \frac{cos ( 2 x ) + 3}{2 ( 1 + cos ( 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{cos ( 2 x ) + 3}{1 + cos ( 2 x )} d x

Rewrite using trig identities

= \frac{1}{2} \cdot \int \frac{3 + cos ( 2 x )}{2 cos ^{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 \frac{1}{2} \cdot \int \frac{3 + cos ( 2 x )}{cos ^{2} ( x )} d x

Expand

\frac{3 + cos ( 2 x )}{cos ^{2} ( x )} : \frac{3}{cos ^{2} ( x )} + \frac{cos ( 2 x )}{cos ^{2} ( x )}

= \frac{1}{2} \cdot \frac{1}{2} \cdot \int \frac{3}{cos ^{2} ( x )} + \frac{cos ( 2 x )}{cos ^{2} ( x )} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} \cdot \frac{1}{2} (\int \frac{3}{cos ^{2} ( x )} d x + \int \frac{cos ( 2 x )}{cos ^{2} ( x )} d x)

\int \frac{3}{cos ^{2} ( x )} d x = 3 tan (x)

\int \frac{cos ( 2 x )}{cos ^{2} ( x )} d x = - tan (x) + 2 x

= \frac{1}{2} \cdot \frac{1}{2} (3 tan (x) - tan (x) + 2 x)

Simplify

\frac{1}{2} \cdot \frac{1}{2} (3 tan (x) - tan (x) + 2 x) : \frac{1}{2} (tan (x) + x)

= \frac{1}{2} (tan (x) + x)

Add a constant to the solution

= \frac{1}{2} (tan (x) + x) + C

\int \frac{1}{x ^{2} + 2 x + 4} d x

\int cos (x) cos (\frac{x}{2}) d x

\int \frac{4 x ^{3} - 3 x}{x ^{4} - 2 x ^{2} + 5} d x

\int \frac{x ^{2} - 1}{x ^{4} + 1} d x

\int \frac{x + 5}{x ^{2} - 6 x + 13} d x

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