Integral of xsin0,5xdx dx

The solution

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$$\int\limits_{0}^{\frac{\pi}{3}} x \sin{\left(0.5 x \right)}\, dx$$

Integral(x*sin(0.5*x), (x, 0, pi/3))

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 $\operatorname{dv}{\left(x \right)} = \sin{\left(0.5 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 0.5 x$ .
  
  Then let $du = 0.5 dx$ and substitute $2.0 du$ :
  
  $\int 2.0 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = 2.0 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 2.0 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 2.0 \cos{\left(0.5 x \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 2.0 \cos{\left(0.5 x \right)}\right)\, dx = - 2.0 \int \cos{\left(0.5 x \right)}\, dx$
1. Let $u = 0.5 x$ .
  
  Then let $du = 0.5 dx$ and substitute $2.0 du$ :
  
  $\int 2.0 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 2.0 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2.0 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2.0 \sin{\left(0.5 x \right)}$
So, the result is: $- 4.0 \sin{\left(0.5 x \right)}$
Add the constant of integration:

$- 2.0 x \cos{\left(0.5 x \right)} + 4.0 \sin{\left(0.5 x \right)}+ \mathrm{constant}$

The answer is:

$- 2.0 x \cos{\left(0.5 x \right)} + 4.0 \sin{\left(0.5 x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
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 | x*sin(0.5*x) dx = C + 4.0*sin(0.5*x) - 2.0*x*cos(0.5*x)
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$$\int x \sin{\left(0.5 x \right)}\, dx = C - 2.0 x \cos{\left(0.5 x \right)} + 4.0 \sin{\left(0.5 x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4.0*sin(0.166666666666667*pi) - 0.666666666666667*pi*cos(0.166666666666667*pi)

$$- 0.666666666666667 \pi \cos{\left(0.166666666666667 \pi \right)} + 4.0 \sin{\left(0.166666666666667 \pi \right)}$$

4.0*sin(0.166666666666667*pi) - 0.666666666666667*pi*cos(0.166666666666667*pi)

$$- 0.666666666666667 \pi \cos{\left(0.166666666666667 \pi \right)} + 4.0 \sin{\left(0.166666666666667 \pi \right)}$$

4.0*sin(0.166666666666667*pi) - 0.666666666666667*pi*cos(0.166666666666667*pi)

Numerical answer [src]

0.186200635765782

0.186200635765782

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

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Identical expressions

Integral of xsin0,5xdx dx

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