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Integral of x(cosx-1/2) dx

The solution

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  pi                    
  --                    
  3                     
   /                    
  |                     
  |  x*(cos(x) - 1/2) dx
  |                     
 /                      
-pi                     
----                    
 3

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

Integral(x*(cos(x) - 1/2), (x, -pi/3, pi/3))

Detail solution

Rewrite the integrand:

$x \left(\cos{\left(x \right)} - \frac{1}{2}\right) = x \cos{\left(x \right)} - \frac{x}{2}$
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 $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 1$ .
  
  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.
2. The integral of sine is negative cosine:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x}{2}\right)\, dx = - \frac{\int x\, dx}{2}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x\, dx = \frac{x^{2}}{2}$
  So, the result is: $- \frac{x^{2}}{4}$
The result is: $- \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                           2                    
 |                           x                     
 | x*(cos(x) - 1/2) dx = C - -- + x*sin(x) + cos(x)
 |                           4                     
/

$$\int x \left(\cos{\left(x \right)} - \frac{1}{2}\right)\, dx = C - \frac{x^{2}}{4} + x \sin{\left(x \right)} + \cos{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

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Integral of x(cosx-1/2) dx

Limits of integration:

The graph:

Piecewise:

The solution