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Integral of d{x}:
Integral of sin^2(3x)
Integral of x^2/2
Integral of tan^2x
Integral of 10^x
Derivative of:
sin^2(3x)
Identical expressions
sin^ two (3x)
sinus of squared (3x)
sinus of to the power of two (3x)
sin2(3x)
sin23x
sin²(3x)
sin to the power of 2(3x)
sin^23x
sin^2(3x)dx

Solving integrals/ sin^2(3x)

Integral of sin^2(3x) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\sin^{2}{\left(3 x \right)} = \frac{1}{2} - \frac{\cos{\left(6 x \right)}}{2}$
Integrate term-by-term:
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\cos{\left(6 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(6 x \right)}\, dx}{2}$
  1. Let $u = 6 x$ .
    
    Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{\cos{\left(u \right)}}{36}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{6}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
      1. The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      So, the result is: $\frac{\sin{\left(u \right)}}{6}$
    Now substitute $u$ back in:
    
    $\frac{\sin{\left(6 x \right)}}{6}$
  So, the result is: $- \frac{\sin{\left(6 x \right)}}{12}$
The result is: $\frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12}$
Add the constant of integration:

$\frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12}+ \mathrm{constant}$

The answer is:

$\frac{x}{2} - \frac{\sin{\left(6 x \right)}}{12}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                                
 |    2               x   sin(6*x)
 | sin (3*x) dx = C + - - --------
 |                    2      12   
/

$${{3\,x-{{\sin \left(6\,x\right)}\over{2}}}\over{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1   cos(3)*sin(3)
- - -------------
2         6

$$-{{\sin 6-6}\over{12}}$$

1   cos(3)*sin(3)
- - -------------
2         6

$$- \frac{\sin{\left(3 \right)} \cos{\left(3 \right)}}{6} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.52328462484991

0.52328462484991

The graph

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

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Identical expressions

Integral of sin^2(3x) dx

Limits of integration:

The graph:

Piecewise:

The solution