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Identical expressions
(sin three x*cos3x)/3
( sinus of 3x multiply by co sinus of e of 3x) divide by 3
( sinus of three x multiply by co sinus of e of 3x) divide by 3
(sin3xcos3x)/3
sin3xcos3x/3
(sin3x*cos3x) divide by 3
(sin3x*cos3x)/3dx

Integral of (sin3x*cos3x)/3 dx

The solution

You have entered [src]

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

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

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

Detail solution

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

$\int \frac{\sin{\left(3 x \right)} \cos{\left(3 x \right)}}{3}\, dx = \frac{\int \sin{\left(3 x \right)} \cos{\left(3 x \right)}\, dx}{3}$
1. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \cos{\left(3 x \right)}$ .
    
    Then let $du = - 3 \sin{\left(3 x \right)} dx$ and substitute $- \frac{du}{3}$ :
    
    $\int \left(- \frac{u}{3}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \frac{\int u\, du}{3}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{6}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(3 x \right)}}{6}$
  Method #2
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\sin{\left(u \right)} \cos{\left(u \right)}}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)} \cos{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)} \cos{\left(u \right)}\, du}{3}$
      
      Let $u = \cos{\left(u \right)}$ .
      
      Then let $du = - \sin{\left(u \right)} du$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(u \right)}}{2}$
      
      So, the result is: $- \frac{\cos^{2}{\left(u \right)}}{6}$
    Now substitute $u$ back in:
    
    $- \frac{\cos^{2}{\left(3 x \right)}}{6}$
  Method #3
  1. Let $u = \sin{\left(3 x \right)}$ .
    
    Then let $du = 3 \cos{\left(3 x \right)} dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{u}{3}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = \frac{\int u\, du}{3}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $\frac{u^{2}}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sin^{2}{\left(3 x \right)}}{6}$
So, the result is: $- \frac{\cos^{2}{\left(3 x \right)}}{18}$
Add the constant of integration:

$- \frac{\cos^{2}{\left(3 x \right)}}{18}+ \mathrm{constant}$

The answer is:

$- \frac{\cos^{2}{\left(3 x \right)}}{18}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                               2     
 | sin(3*x)*cos(3*x)          cos (3*x)
 | ----------------- dx = C - ---------
 |         3                      18   
 |                                     
/

$$\int \frac{\sin{\left(3 x \right)} \cos{\left(3 x \right)}}{3}\, dx = C - \frac{\cos^{2}{\left(3 x \right)}}{18}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   2   
sin (3)
-------
   18

$$\frac{\sin^{2}{\left(3 \right)}}{18}$$

   2   
sin (3)
-------
   18

$$\frac{\sin^{2}{\left(3 \right)}}{18}$$

sin(3)^2/18

Numerical answer [src]

0.00110638092637872

0.00110638092637872

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

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Identical expressions

Integral of (sin3x*cos3x)/3 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3