Integral of 2*sin(3x) dx

You have entered [src]

 pi              
 --              
 2               
  /              
 |               
 |  2*sin(3*x) dx
 |               
/                
0

\int\limits_{0}^{\frac{\pi}{2}} 2 \sin{\left(3 x \right)}\, dx

Integral(2*sin(3*x), (x, 0, pi/2))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 2 \sin{\left(3 x \right)}\, dx = 2 \int \sin{\left(3 x \right)}\, dx$
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
  
  $\int \frac{\sin{\left(u \right)}}{3}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 x \right)}}{3}$
So, the result is: $- \frac{2 \cos{\left(3 x \right)}}{3}$
Add the constant of integration:

$- \frac{2 \cos{\left(3 x \right)}}{3}+ \mathrm{constant}$

The answer is:

- \frac{2 \cos{\left(3 x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                              
 |                     2*cos(3*x)
 | 2*sin(3*x) dx = C - ----------
 |                         3     
/

\int 2 \sin{\left(3 x \right)}\, dx = C - \frac{2 \cos{\left(3 x \right)}}{3}

Simplify

The graph

Plot the graph f(x)

The answer [src]

2/3

\frac{2}{3}

2/3

\frac{2}{3}

2/3

Numerical answer [src]

0.666666666666667

0.666666666666667

Use the examples entering the upper and lower limits of integration.