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Integral of d{x}:
Integral of 1/(2+cos(x))
Integral of √((1-x)/(1+x))
Integral of cos(x)*exp(x)
Integral of (3x-2)^2
Identical expressions
x*a*sin(x/a)
x multiply by a multiply by sinus of (x divide by a)
xasin(x/a)
xasinx/a
x*a*sin(x divide by a)
x*a*sin(x/a)dx

Integral of xasin(x/a) dx

The solution

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  1              
  /              
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 |         /x\   
 |  x*a*sin|-| dx
 |         \a/   
 |               
/                
0

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

Integral((x*a)*sin(x/a), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = a x$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{a} \right)}$ .

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

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                           
 |                                            
 |        /x\           3    /x\      2    /x\
 | x*a*sin|-| dx = C + a *sin|-| - x*a *cos|-|
 |        \a/                \a/           \a/
 |                                            
/

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

Simplify

The answer [src]

  / 2    /1\        /1\\
a*|a *sin|-| - a*cos|-||
  \      \a/        \a//

$$a \left(a^{2} \sin{\left(\frac{1}{a} \right)} - a \cos{\left(\frac{1}{a} \right)}\right)$$

  / 2    /1\        /1\\
a*|a *sin|-| - a*cos|-||
  \      \a/        \a//

$$a \left(a^{2} \sin{\left(\frac{1}{a} \right)} - a \cos{\left(\frac{1}{a} \right)}\right)$$

a*(a^2*sin(1/a) - a*cos(1/a))

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

Other calculators

How to use it?

Identical expressions

Integral of xasin(x/a) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Integral of x*a*sin(x/a) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of xasin(x/a) dx