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

The integral of 1/2 (cos(x))^2 is 1/4 (x+1/2 sin(2x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{1}{4} (x + \frac{1}{2} sin (2 x)) + C

Solution steps

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

Simplify

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

Rewrite using trig identities

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

\int 1 d x = x

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

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

Simplify

\frac{1}{2} \cdot \frac{1}{2} (x + \frac{1}{2} sin (2 x)) : \frac{1}{4} (x + \frac{1}{2} sin (2 x))

= \frac{1}{4} (x + \frac{1}{2} sin (2 x))

Add a constant to the solution

= \frac{1}{4} (x + \frac{1}{2} sin (2 x)) + C

f (x) = \frac{40}{x}, (1, 40)

\int (\frac{x + 1}{x}) d x

\int \frac{1}{2} x^{5} d x

x \to \frac{π}{2} + lim (2 e^{t a n (x)})

f (x) = 3 x^{4} - 8 x^{2}, (- 1, 5)

What is the integral of 1/2 (cos(x))^2 ?
The integral of 1/2 (cos(x))^2 is 1/4 (x+1/2 sin(2x))+C