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Identical expressions
x^ two *sin(x×pi)
x squared multiply by sinus of (x× Pi )
x to the power of two multiply by sinus of (x× Pi )
x2*sin(x×pi)
x2*sinx×pi
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x to the power of 2*sin(x×pi)
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x2sin(x×pi)
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x^2sinx×pi
x^2*sin(x×pi)dx

Solving integrals/ x^2*sin(x×pi)

Integral of x^2*sin(x×pi) dx

The solution

You have entered [src]

 pi                
  /                
 |                 
 |   2             
 |  x *sin(x*pi) dx
 |                 
/                  
0

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

Simplify

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(\pi x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

To find $v{\left(x \right)}$ :
1. Let $u = \pi x$ .
  
  Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
  
  $\int \frac{\sin{\left(u \right)}}{\pi^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{\pi}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}$
    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:
  
  $- \frac{\cos{\left(\pi x \right)}}{\pi}$
Now evaluate the sub-integral.
Use integration by parts:

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

Let $u{\left(x \right)} = - \frac{2 x}{\pi}$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(\pi x \right)}$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{2}{\pi}$ .

To find $v{\left(x \right)}$ :
1. Let $u = \pi x$ .
  
  Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
  
  $\int \frac{\cos{\left(u \right)}}{\pi^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\cos{\left(u \right)}}{\pi}\, du = \frac{\int \cos{\left(u \right)}\, du}{\pi}$
    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:

$\int \left(- \frac{2 \sin{\left(\pi x \right)}}{\pi^{2}}\right)\, dx = - \frac{2 \int \sin{\left(\pi x \right)}\, dx}{\pi^{2}}$
1. Let $u = \pi x$ .
  
  Then let $du = \pi dx$ and substitute $\frac{du}{\pi}$ :
  
  $\int \frac{\sin{\left(u \right)}}{\pi^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{\pi}\, du = \frac{\int \sin{\left(u \right)}\, du}{\pi}$
    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: $\frac{2 \cos{\left(\pi x \right)}}{\pi^{3}}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

                           /  2\        /  2\
   2          /  2\   2*sin\pi /   2*cos\pi /
- --- - pi*cos\pi / + ---------- + ----------
    3                     pi            3    
  pi                                  pi

$$\frac{2 \sin{\left(\pi^{2} \right)}}{\pi} - \frac{2}{\pi^{3}} + \frac{2 \cos{\left(\pi^{2} \right)}}{\pi^{3}} - \pi \cos{\left(\pi^{2} \right)}$$

                           /  2\        /  2\
   2          /  2\   2*sin\pi /   2*cos\pi /
- --- - pi*cos\pi / + ---------- + ----------
    3                     pi            3    
  pi                                  pi

$$\frac{2 \sin{\left(\pi^{2} \right)}}{\pi} - \frac{2}{\pi^{3}} + \frac{2 \cos{\left(\pi^{2} \right)}}{\pi^{3}} - \pi \cos{\left(\pi^{2} \right)}$$

Simplify

Numerical answer [src]

2.43920239380403

2.43920239380403

The graph

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

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Identical expressions

Integral of x^2*sin(x×pi) dx

Limits of integration:

The graph:

Piecewise:

The solution