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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

3 x - 4 cos (x) - sin (2 x) + C

Solution steps

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

Expand

(1 + 2 sin (x))^{2} : 1 + 4 sin (x) + 4 sin^{2} (x)

= \int 1 + 4 sin (x) + 4 sin^{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

= \int 1 d x + \int 4 sin (x) d x + \int 4 sin^{2} (x) d x

\int 1 d x = x

\int 4 sin (x) d x = - 4 cos (x)

\int 4 sin^{2} (x) d x = 2 x - sin (2 x)

= x - 4 cos (x) + 2 x - sin (2 x)

Simplify

= 3 x - 4 cos (x) - sin (2 x)

Add a constant to the solution

= 3 x - 4 cos (x) - sin (2 x) + C

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