Integral of xcos(pi*x) dx

The solution

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

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

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

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\pi x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

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

$\int \frac{\sin{\left(\pi x \right)}}{\pi}\, dx = \frac{\int \sin{\left(\pi x \right)}\, dx}{\pi}$
1. Let $u = \pi x$ .
  
  Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
  
  $\int \frac{\sin{\left(u \right)}}{\pi}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{\pi}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(\pi x \right)}}{\pi}$
So, the result is: $- \frac{\cos{\left(\pi x \right)}}{\pi^{2}}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2 
---
  2
pi

$$- \frac{2}{\pi^{2}}$$

-2 
---
  2
pi

$$- \frac{2}{\pi^{2}}$$

-2/pi^2

Numerical answer [src]

-0.202642367284676

-0.202642367284676

The graph

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

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Identical expressions

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Integral of xcos(pi*x) dx

Limits of integration:

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Piecewise:

The solution