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Similar expressions
(x+9)•sin*x/2dx

Integral of (x-9)•sin*x/2dx dx

The solution

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  1                  
  /                  
 |                   
 |  (x - 9)*sin(x)   
 |  -------------- dx
 |        2          
 |                   
/                    
0

$$\int\limits_{0}^{1} \frac{\left(x - 9\right) \sin{\left(x \right)}}{2}\, dx$$

Integral(((x - 9)*sin(x))/2, (x, 0, 1))

Detail solution

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

$\int \frac{\left(x - 9\right) \sin{\left(x \right)}}{2}\, dx = \frac{\int \left(x - 9\right) \sin{\left(x \right)}\, dx}{2}$
1. There are multiple ways to do this integral.
  Method #1
  1. Rewrite the integrand:
    
    $\left(x - 9\right) \sin{\left(x \right)} = x \sin{\left(x \right)} - 9 \sin{\left(x \right)}$
  2. Integrate term-by-term:
    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 $v{\left(x \right)}$ :
      
      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:
      
      $\int \left(- \cos{\left(x \right)}\right)\, dx = - \int \cos{\left(x \right)}\, dx$
      
      The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
      
      So, the result is: $- \sin{\left(x \right)}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 9 \sin{\left(x \right)}\right)\, dx = - 9 \int \sin{\left(x \right)}\, dx$
      
      The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
      
      So, the result is: $9 \cos{\left(x \right)}$
    The result is: $- x \cos{\left(x \right)} + \sin{\left(x \right)} + 9 \cos{\left(x \right)}$
  Method #2
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = x - 9$ 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: $- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{9 \cos{\left(x \right)}}{2}$
Add the constant of integration:

$- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{9 \cos{\left(x \right)}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{9 \cos{\left(x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
 |                                                     
 | (x - 9)*sin(x)          sin(x)   9*cos(x)   x*cos(x)
 | -------------- dx = C + ------ + -------- - --------
 |       2                   2         2          2    
 |                                                     
/

$$\int \frac{\left(x - 9\right) \sin{\left(x \right)}}{2}\, dx = C - \frac{x \cos{\left(x \right)}}{2} + \frac{\sin{\left(x \right)}}{2} + \frac{9 \cos{\left(x \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$- \frac{9}{2} + \frac{\sin{\left(1 \right)}}{2} + 4 \cos{\left(1 \right)}$$

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

$$- \frac{9}{2} + \frac{\sin{\left(1 \right)}}{2} + 4 \cos{\left(1 \right)}$$

-9/2 + sin(1)/2 + 4*cos(1)

Numerical answer [src]

-1.91805528412349

-1.91805528412349

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

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Identical expressions

Similar expressions

Integral of (x-9)•sin*x/2dx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2