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cos(2pix)cos(pix)
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cos2pixcospix
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Solving integrals/ cos(2pix)cos(pix)

Integral of cos(2pix)cos(pix) dx

The solution

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

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

Integral(cos(2*pi*x)*cos(pi*x), (x, 0, 1))

Detail solution

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

$\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}$
Add the constant of integration:

$\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}+ \mathrm{constant}$

The answer is:

$\frac{\left(\cos{\left(2 \pi x \right)} + 2\right) \sin{\left(\pi x \right)}}{3 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

$${{\sin \left(3\,\pi\right)+3\,\sin \pi}\over{6\,\pi}}$$

$$0$$

Упростить

Numerical answer [src]

-1.02704285974057e-22

-1.02704285974057e-22

The graph

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

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Identical expressions

Integral of cos(2pix)cos(pix) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2