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

The integral of (cos(x)+1)^2 is 3/2 x+1/4 sin(2x)+2sin(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C

Solution steps

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

Expand

(cos (x) + 1)^{2} : cos^{2} (x) + 2 cos (x) + 1

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

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

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

\int 2 cos (x) d x = 2 sin (x)

\int 1 d x = x

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

Simplify

\frac{1}{2} (x + \frac{1}{2} sin (2 x)) + 2 sin (x) + x : \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x)

Add a constant to the solution

= \frac{3}{2} x + \frac{1}{4} sin (2 x) + 2 sin (x) + C

\frac{d P}{d t} = Pt, P (1) = 2

\int 2 x + x^{2} d x

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

y^{^{'}} = x + y, y (0) = 8

What is the integral of (cos(x)+1)^2 ?
The integral of (cos(x)+1)^2 is 3/2 x+1/4 sin(2x)+2sin(x)+C