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Solving integrals/ (x^2-3x+2)sinx

Integral of (x^2-3x+2)sinx dx

The solution

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0

$$\int\limits_{0}^{1} \left(\left(x^{2} - 3 x\right) + 2\right) \sin{\left(x \right)}\, dx$$

Integral((x^2 - 3*x + 2)*sin(x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(\left(x^{2} - 3 x\right) + 2\right) \sin{\left(x \right)} = x^{2} \sin{\left(x \right)} - 3 x \sin{\left(x \right)} + 2 \sin{\left(x \right)}$
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 $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = 2 x$ .
    
    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. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(x \right)} = - 2 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
    
    Then $\operatorname{du}{\left(x \right)} = -2$ .
    
    To find $v{\left(x \right)}$ :
    1. The integral of cosine is sine:
      
      $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
    Now evaluate the sub-integral.
  3. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 \sin{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)}\, dx$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    So, the result is: $2 \cos{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 3 x \sin{\left(x \right)}\right)\, dx = - 3 \int x \sin{\left(x \right)}\, dx$
    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: $- \sin{\left(x \right)}$
    So, the result is: $3 x \cos{\left(x \right)} - 3 \sin{\left(x \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 2 \sin{\left(x \right)}\, dx = 2 \int \sin{\left(x \right)}\, dx$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
    So, the result is: $- 2 \cos{\left(x \right)}$
  The result is: $- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 3 \sin{\left(x \right)}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = x^{2} - 3 x + 2$ and let $\operatorname{dv}{\left(x \right)} = \sin{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 2 x - 3$ .
  
  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. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 3 - 2 x$ and let $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = -2$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of cosine is sine:
    
    $\int \cos{\left(x \right)}\, dx = \sin{\left(x \right)}$
  Now evaluate the sub-integral.
3. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 2 \sin{\left(x \right)}\right)\, dx = - 2 \int \sin{\left(x \right)}\, dx$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $2 \cos{\left(x \right)}$
Add the constant of integration:

$- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 3 \sin{\left(x \right)}+ \mathrm{constant}$

The answer is:

$- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 3 \sin{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                             
 |                                                                              
 | / 2          \                             2                                 
 | \x  - 3*x + 2/*sin(x) dx = C - 3*sin(x) - x *cos(x) + 2*x*sin(x) + 3*x*cos(x)
 |                                                                              
/

$$\int \left(\left(x^{2} - 3 x\right) + 2\right) \sin{\left(x \right)}\, dx = C - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 3 x \cos{\left(x \right)} - 3 \sin{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

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

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

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

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

Numerical answer [src]

0.239133626928383

0.239133626928383

The graph

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

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Identical expressions

Similar expressions

Integral of (x^2-3x+2)sinx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2