Integral of (xcosx/2) dx

The solution

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  1            
  /            
 |             
 |  x*cos(x)   
 |  -------- dx
 |     2       
 |             
/              
0

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

Integral(x*cos(x)/2, (x, 0, 1))

Detail solution

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

$\int \frac{x \cos{\left(x \right)}}{2}\, dx = \frac{\int x \cos{\left(x \right)}\, dx}{2}$
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 $\operatorname{dv}{\left(x \right)} = \cos{\left(x \right)}$ .
  
  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:
  
  $\int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)}$
So, the result is: $\frac{x \sin{\left(x \right)}}{2} + \frac{\cos{\left(x \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$${{x\,\sin x+\cos x}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

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

$${{\sin 1+\cos 1-1}\over{2}}$$

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

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

Simplify

Numerical answer [src]

0.190886645338018

0.190886645338018

The graph

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

Other calculators

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Identical expressions

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Integral of (xcosx/2) dx

Limits of integration:

The graph:

Piecewise:

The solution