Integral of cos²3x dx

The solution

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  1             
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 |     2        
 |  cos (3*x) dx
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0

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

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

Detail solution

Rewrite the integrand:

$\cos^{2}{\left(3 x \right)} = \frac{\cos{\left(6 x \right)}}{2} + \frac{1}{2}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cos{\left(6 x \right)}}{2}\, dx = \frac{\int \cos{\left(6 x \right)}\, dx}{2}$
  1. Let $u = 6 x$ .
    
    Then let $du = 6 dx$ and substitute $\frac{du}{6}$ :
    
    $\int \frac{\cos{\left(u \right)}}{6}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{6}$
      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: $\frac{\sin{\left(6 x \right)}}{12}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{1}{2}\, dx = \frac{x}{2}$
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)
 | cos (3*x) dx = C + - + --------
 |                    2      12   
/

$$\int \cos^{2}{\left(3 x \right)}\, dx = C + \frac{x}{2} + \frac{\sin{\left(6 x \right)}}{12}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

1/2 + cos(3)*sin(3)/6

Numerical answer [src]

0.476715375150089

0.476715375150089

The graph

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

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Integral of cos²3x dx

Limits of integration:

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Piecewise:

The solution