Integral of sin3x/2 dx

The solution

You have entered [src]

  1            
  /            
 |             
 |  sin(3*x)   
 |  -------- dx
 |     2       
 |             
/              
0

$$\int\limits_{0}^{1} \frac{\sin{\left(3 x \right)}}{2}\, dx$$

Integral(sin(3*x)/2, (x, 0, 1))

Detail solution

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

$\int \frac{\sin{\left(3 x \right)}}{2}\, dx = \frac{\int \sin{\left(3 x \right)}\, dx}{2}$
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: $- \frac{\cos{\left(3 x \right)}}{6}$
Add the constant of integration:

$- \frac{\cos{\left(3 x \right)}}{6}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 x \right)}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          
 |                           
 | sin(3*x)          cos(3*x)
 | -------- dx = C - --------
 |    2                 6    
 |                           
/

$$\int \frac{\sin{\left(3 x \right)}}{2}\, dx = C - \frac{\cos{\left(3 x \right)}}{6}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(3)
- - ------
6     6

$$\frac{1}{6} - \frac{\cos{\left(3 \right)}}{6}$$

1   cos(3)
- - ------
6     6

$$\frac{1}{6} - \frac{\cos{\left(3 \right)}}{6}$$

1/6 - cos(3)/6

Numerical answer [src]

0.331665416100074

0.331665416100074

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of sin3x/2 dx

Limits of integration:

The graph:

Piecewise:

The solution