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Identical expressions
xcos(x/ five)
x co sinus of e of (x divide by 5)
x co sinus of e of (x divide by five)
xcosx/5
xcos(x divide by 5)
xcos(x/5)dx

Solving integrals/ xcos(x/5)

Integral of xcos(x/5) dx

The solution

You have entered [src]

  1            
  /            
 |             
 |       /x\   
 |  x*cos|-| dx
 |       \5/   
 |             
/              
0

$$\int\limits_{0}^{1} x \cos{\left(\frac{x}{5} \right)}\, dx$$

Simplify

Detail solution

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 $\operatorname{dv}{\left(x \right)} = \cos{\left(\frac{x}{5} \right)}$ .

Then $\operatorname{du}{\left(x \right)} = 1$ .

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{5}$ .
  
  Then let $du = \frac{dx}{5}$ and substitute $5 du$ :
  
  $\int 25 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5 \cos{\left(u \right)}\, du = 5 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $5 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $5 \sin{\left(\frac{x}{5} \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int 5 \sin{\left(\frac{x}{5} \right)}\, dx = 5 \int \sin{\left(\frac{x}{5} \right)}\, dx$
1. Let $u = \frac{x}{5}$ .
  
  Then let $du = \frac{dx}{5}$ and substitute $5 du$ :
  
  $\int 25 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 5 \sin{\left(u \right)}\, du = 5 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 5 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 5 \cos{\left(\frac{x}{5} \right)}$
So, the result is: $- 25 \cos{\left(\frac{x}{5} \right)}$
Add the constant of integration:

$5 x \sin{\left(\frac{x}{5} \right)} + 25 \cos{\left(\frac{x}{5} \right)}+ \mathrm{constant}$

The answer is:

$5 x \sin{\left(\frac{x}{5} \right)} + 25 \cos{\left(\frac{x}{5} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                        
 |                                         
 |      /x\                /x\          /x\
 | x*cos|-| dx = C + 25*cos|-| + 5*x*sin|-|
 |      \5/                \5/          \5/
 |                                         
/

$$25\,\left({{\sin \left({{x}\over{5}}\right)\,x}\over{5}}+\cos \left({{x}\over{5}}\right)\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-25 + 5*sin(1/5) + 25*cos(1/5)

$$5\,\sin \left({{1}\over{5}}\right)+25\,\cos \left({{1}\over{5}} \right)-25$$

-25 + 5*sin(1/5) + 25*cos(1/5)

$$-25 + 5 \sin{\left(\frac{1}{5} \right)} + 25 \cos{\left(\frac{1}{5} \right)}$$

Simplify

Numerical answer [src]

0.495011100006347

0.495011100006347

The graph

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

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Identical expressions

Integral of xcos(x/5) dx

Limits of integration:

The graph:

Piecewise:

The solution