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x*sin(two *x-pi/ two)/ two -x*sin(x)/ two
x multiply by sinus of (2 multiply by x minus Pi divide by 2) divide by 2 minus x multiply by sinus of (x) divide by 2
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xsin(2x-pi/2)/2-xsin(x)/2
xsin2x-pi/2/2-xsinx/2
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x*sin(2*x-pi/2)/2-x*sin(x)/2dx
Similar expressions
x*sin(2*x+pi/2)/2-x*sin(x)/2
x*sin(2*x-pi/2)/2+x*sin(x)/2
x*sin(2*x-pi/2)/2-x*sinx/2

Solving integrals/ x*sin(2*x-pi/2)/2-x*sin(x)/2

Integral of xsin(2x-pi/2)/2-x*sin(x)/2 dx

The solution

You have entered [src]

  pi                                
  --                                
  2                                 
   /                                
  |                                 
  |  /     /      pi\           \   
  |  |x*sin|2*x - --|           |   
  |  |     \      2 /   x*sin(x)|   
  |  |--------------- - --------| dx
  |  \       2             2    /   
  |                                 
 /                                  
-pi                                 
----                                
 2

$$\int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \left(- \frac{x \sin{\left(x \right)}}{2} + \frac{x \sin{\left(2 x - \frac{\pi}{2} \right)}}{2}\right)\, dx$$

Integral((x*sin(2*x - pi/2))/2 - x*sin(x)/2, (x, -pi/2, pi/2))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x \sin{\left(x \right)}}{2}\right)\, dx = - \frac{\int x \sin{\left(x \right)}\, dx}{2}$
  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: $\frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x \sin{\left(2 x - \frac{\pi}{2} \right)}}{2}\, dx = \frac{\int x \sin{\left(2 x - \frac{\pi}{2} \right)}\, dx}{2}$
  1. There are multiple ways to do this integral.
    Method #1
    1. Rewrite the integrand:
      
      $x \sin{\left(2 x - \frac{\pi}{2} \right)} = - x \cos{\left(2 x \right)}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x \cos{\left(2 x \right)}\right)\, dx = - \int x \cos{\left(2 x \right)}\, dx$
      
      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 $\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(2 x \right)}}{2}\, dx = \frac{\int \sin{\left(2 x \right)}\, dx}{2}$
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\sin{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(2 x \right)}}{2}$
      
      Method #2
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
      
      Let $u = \cos{\left(x \right)}$ .
      
      Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :
      
      $\int \left(- u\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos^{2}{\left(x \right)}}{2}$
      
      So, the result is: $- \cos^{2}{\left(x \right)}$
      
      So, the result is: $- \frac{\cos{\left(2 x \right)}}{4}$
      
      So, the result is: $- \frac{x \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{4}$
    Method #2
    1. Rewrite the integrand:
      
      $x \sin{\left(2 x - \frac{\pi}{2} \right)} = - x \cos{\left(2 x \right)}$
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x \cos{\left(2 x \right)}\right)\, dx = - \int x \cos{\left(2 x \right)}\, dx$
      
      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 $\operatorname{dv}{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
      
      So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
      
      Now evaluate the sub-integral.
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\sin{\left(2 x \right)}}{2}\, dx = \frac{\int \sin{\left(2 x \right)}\, dx}{2}$
      
      Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\sin{\left(u \right)}}{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{2}$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
      
      So, the result is: $- \frac{\cos{\left(u \right)}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\cos{\left(2 x \right)}}{2}$
      
      So, the result is: $- \frac{\cos{\left(2 x \right)}}{4}$
      
      So, the result is: $- \frac{x \sin{\left(2 x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{4}$
  So, the result is: $- \frac{x \sin{\left(2 x \right)}}{4} - \frac{\cos{\left(2 x \right)}}{8}$
The result is: $- \frac{x \sin{\left(2 x \right)}}{4} + \frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{8}$
Add the constant of integration:

$- \frac{x \sin{\left(2 x \right)}}{4} + \frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer is:

$- \frac{x \sin{\left(2 x \right)}}{4} + \frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                               
 |                                                                                
 | /     /      pi\           \                                                   
 | |x*sin|2*x - --|           |                                                   
 | |     \      2 /   x*sin(x)|          sin(x)   cos(2*x)   x*cos(x)   x*sin(2*x)
 | |--------------- - --------| dx = C - ------ - -------- + -------- - ----------
 | \       2             2    /            2         8          2           4     
 |                                                                                
/

$$\int \left(- \frac{x \sin{\left(x \right)}}{2} + \frac{x \sin{\left(2 x - \frac{\pi}{2} \right)}}{2}\right)\, dx = C - \frac{x \sin{\left(2 x \right)}}{4} + \frac{x \cos{\left(x \right)}}{2} - \frac{\sin{\left(x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1

$$-1$$

-1

$$-1$$

-1

Numerical answer [src]

-1.0

-1.0

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of xsin(2x-pi/2)/2-x*sin(x)/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x*sin(2*x-pi/2)/2-x*sin(x)/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Integral of xsin(2x-pi/2)/2-x*sin(x)/2 dx