Mister Exam

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(two *x- three)*(sinx/ two)
(2 multiply by x minus 3) multiply by ( sinus of x divide by 2)
(two multiply by x minus three) multiply by ( sinus of x divide by two)
(2x-3)(sinx/2)
2x-3sinx/2
(2*x-3)*(sinx divide by 2)
(2*x-3)*(sinx/2)dx
Similar expressions
(2x-3)*(sinx/2)
(2*x+3)*(sinx/2)

Integral of (2x-3)(sinx/2) dx

The solution

You have entered [src]

  1                    
  /                    
 |                     
 |            sin(x)   
 |  (2*x - 3)*------ dx
 |              2      
 |                     
/                      
0

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

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

Detail solution

Rewrite the integrand:

$\frac{\sin{\left(x \right)}}{2} \left(2 x - 3\right) = x \sin{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{2}$
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 $v{\left(x \right)}$ :
  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: $- \sin{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{3 \sin{\left(x \right)}}{2}\right)\, dx = - \frac{3 \int \sin{\left(x \right)}\, dx}{2}$
  1. The integral of sine is negative cosine:
    
    $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
  So, the result is: $\frac{3 \cos{\left(x \right)}}{2}$
The result is: $- x \cos{\left(x \right)} + \sin{\left(x \right)} + \frac{3 \cos{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$$- \frac{3}{2} + \frac{\cos{\left(1 \right)}}{2} + \sin{\left(1 \right)}$$

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

$$- \frac{3}{2} + \frac{\cos{\left(1 \right)}}{2} + \sin{\left(1 \right)}$$

-3/2 + cos(1)/2 + sin(1)

Numerical answer [src]

-0.388377862258034

-0.388377862258034

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

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Identical expressions

Similar expressions

Integral of (2x-3)(sinx/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (2*x-3)*(sinx/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (2x-3)(sinx/2) dx