Mister Exam

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Identical expressions
xsin(x/ three)*dx
x sinus of (x divide by 3) multiply by dx
x sinus of (x divide by three) multiply by dx
xsin(x/3)dx
xsinx/3dx
xsin(x divide by 3)*dx

Integral of xsin(x/3)*dx dx

The solution

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

Integral(x*sin(x/3), (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 $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{3} \right)}$ .

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

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

$\int \left(- 3 \cos{\left(\frac{x}{3} \right)}\right)\, dx = - 3 \int \cos{\left(\frac{x}{3} \right)}\, dx$
1. Let $u = \frac{x}{3}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 3 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 3 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $3 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $3 \sin{\left(\frac{x}{3} \right)}$
So, the result is: $- 9 \sin{\left(\frac{x}{3} \right)}$
Add the constant of integration:

$- 3 x \cos{\left(\frac{x}{3} \right)} + 9 \sin{\left(\frac{x}{3} \right)}+ \mathrm{constant}$

The answer is:

$- 3 x \cos{\left(\frac{x}{3} \right)} + 9 \sin{\left(\frac{x}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
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 |      /x\               /x\          /x\
 | x*sin|-| dx = C + 9*sin|-| - 3*x*cos|-|
 |      \3/               \3/          \3/
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/

$$\int x \sin{\left(\frac{x}{3} \right)}\, dx = C - 3 x \cos{\left(\frac{x}{3} \right)} + 9 \sin{\left(\frac{x}{3} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-3*cos(1/3) + 9*sin(1/3)

$$- 3 \cos{\left(\frac{1}{3} \right)} + 9 \sin{\left(\frac{1}{3} \right)}$$

-3*cos(1/3) + 9*sin(1/3)

$$- 3 \cos{\left(\frac{1}{3} \right)} + 9 \sin{\left(\frac{1}{3} \right)}$$

-3*cos(1/3) + 9*sin(1/3)

Numerical answer [src]

0.109881432221157

0.109881432221157

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

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Identical expressions

Integral of xsin(x/3)*dx dx

Limits of integration:

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Piecewise:

The solution