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Identical expressions
sin three pix/3^ two
sinus of 3 Pi x divide by 3 squared
sinus of three Pi x divide by 3 to the power of two
sin3pix/32
sin3pix/3²
sin3pix/3 to the power of 2
sin3pix divide by 3^2
sin3pix/3^2dx

Integral of sin3pix/3^2 dx

The solution

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  1               
  /               
 |                
 |  sin(3*pi*x)   
 |  ----------- dx
 |       9        
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{\sin{\left(3 \pi x \right)}}{9}\, dx$$

Integral(sin((3*pi)*x)/9, (x, 0, 1))

Detail solution

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

$\int \frac{\sin{\left(3 \pi x \right)}}{9}\, dx = \frac{\int \sin{\left(3 \pi x \right)}\, dx}{9}$
1. Let $u = 3 \pi x$ .
  
  Then let $du = 3 \pi dx$ and substitute $\frac{du}{3 \pi}$ :
  
  $\int \frac{\sin{\left(u \right)}}{3 \pi}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{3 \pi}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{3 \pi}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(3 \pi x \right)}}{3 \pi}$
So, the result is: $- \frac{\cos{\left(3 \pi x \right)}}{27 \pi}$
Now simplify:

$- \frac{\cos{\left(3 \pi x \right)}}{27 \pi}$
Add the constant of integration:

$- \frac{\cos{\left(3 \pi x \right)}}{27 \pi}+ \mathrm{constant}$

The answer is:

$- \frac{\cos{\left(3 \pi x \right)}}{27 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                
 |                                 
 | sin(3*pi*x)          cos(3*pi*x)
 | ----------- dx = C - -----------
 |      9                  27*pi   
 |                                 
/

$$\int \frac{\sin{\left(3 \pi x \right)}}{9}\, dx = C - \frac{\cos{\left(3 \pi x \right)}}{27 \pi}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  2  
-----
27*pi

$$\frac{2}{27 \pi}$$

  2  
-----
27*pi

$$\frac{2}{27 \pi}$$

2/(27*pi)

Numerical answer [src]

0.0235785100876882

0.0235785100876882

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

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Identical expressions

Integral of sin3pix/3^2 dx

Limits of integration:

The graph:

Piecewise:

The solution