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Integral of d{x}:
Integral of x*2^x
Integral of x*sin(x/2)
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Integral of x^2*a^x
Derivative of:
x*sin(x/2)
Limit of the function:
x*sin(x/2)
Identical expressions
x*sin(x/ two)
x multiply by sinus of (x divide by 2)
x multiply by sinus of (x divide by two)
xsin(x/2)
xsinx/2
x*sin(x divide by 2)
x*sin(x/2)dx
Similar expressions
(2-x)sinx/2
exp(cos(x)*exp(-x)/2-exp(-x)*sin(x)/2)
(x*sin(x)/2)*dx
sin(5x)sin(x/2)
(e^(-x))*(sin(x/2))

Solving integrals/ x*sin(x/2)

Integral of x*sin(x/2) dx

The solution

You have entered [src]

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$$\int\limits_{0}^{1} x \sin{\left(\frac{x}{2} \right)}\, dx$$

Integral(x*sin(x/2), (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 $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{2} \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 4 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(u \right)}\, du = 2 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 2 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 2 \cos{\left(\frac{x}{2} \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 2 \cos{\left(\frac{x}{2} \right)}\right)\, dx = - 2 \int \cos{\left(\frac{x}{2} \right)}\, dx$
1. Let $u = \frac{x}{2}$ .
  
  Then let $du = \frac{dx}{2}$ and substitute $2 du$ :
  
  $\int 4 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \cos{\left(u \right)}\, du = 2 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $2 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $2 \sin{\left(\frac{x}{2} \right)}$
So, the result is: $- 4 \sin{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$- 2 x \cos{\left(\frac{x}{2} \right)} + 4 \sin{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$- 2 x \cos{\left(\frac{x}{2} \right)} + 4 \sin{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                       
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 |      /x\               /x\          /x\
 | x*sin|-| dx = C + 4*sin|-| - 2*x*cos|-|
 |      \2/               \2/          \2/
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/

$$\int x \sin{\left(\frac{x}{2} \right)}\, dx = C - 2 x \cos{\left(\frac{x}{2} \right)} + 4 \sin{\left(\frac{x}{2} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2*cos(1/2) + 4*sin(1/2)

$$- 2 \cos{\left(\frac{1}{2} \right)} + 4 \sin{\left(\frac{1}{2} \right)}$$

-2*cos(1/2) + 4*sin(1/2)

$$- 2 \cos{\left(\frac{1}{2} \right)} + 4 \sin{\left(\frac{1}{2} \right)}$$

Упростить

Numerical answer [src]

0.162537030636067

0.162537030636067

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x*sin(x/2) dx

Limits of integration:

The graph:

Piecewise:

The solution