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

The integral of 2cos(2x+pi/2) is 2xcos(2x+pi/2)+2xsin(2x)+cos(2x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of 2cos(2x+pi/2)

\int 2 cos (2 x + \frac{π}{2}) d x

2 x cos (2 x + \frac{π}{2}) + 2 x sin (2 x) + cos (2 x) + C

Solution steps

\int 2 cos (2 x + \frac{π}{2}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Apply Integration By Parts

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

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

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

Simplify

2 (x cos (2 x + \frac{π}{2}) - (- x sin (2 x) - \frac{1}{2} cos (2 x))) : 2 x cos (2 x + \frac{π}{2}) + 2 x sin (2 x) + cos (2 x)

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

Add a constant to the solution

= 2 x cos (2 x + \frac{π}{2}) + 2 x sin (2 x) + cos (2 x) + C

What is the integral of 2cos(2x+pi/2) ?
The integral of 2cos(2x+pi/2) is 2xcos(2x+pi/2)+2xsin(2x)+cos(2x)+C