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Identical expressions
x*sinx/3dx
x multiply by sinus of x divide by 3dx
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Similar expressions
xsin(x/3)*dx

Integral of x*sinx/3dx dx

The solution

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  1            
  /            
 |             
 |  x*sin(x)   
 |  -------- dx
 |     3       
 |             
/              
0

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

Integral((x*sin(x))/3, (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{x \sin{\left(x \right)}}{3}\, dx = \frac{\int x \sin{\left(x \right)}\, dx}{3}$
1. 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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  So, the result is: $- \sin{\left(x \right)}$
So, the result is: $- \frac{x \cos{\left(x \right)}}{3} + \frac{\sin{\left(x \right)}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  cos(1)   sin(1)
- ------ + ------
    3        3

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

  cos(1)   sin(1)
- ------ + ------
    3        3

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

-cos(1)/3 + sin(1)/3

Numerical answer [src]

0.100389559646586

0.100389559646586

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

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Identical expressions

Similar expressions

Integral of x*sinx/3dx dx

Limits of integration:

The graph:

Piecewise:

The solution