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Integral of xcos((x-pi)/3) dx

The solution

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  1                 
  /                 
 |                  
 |       /x - pi\   
 |  x*cos|------| dx
 |       \  3   /   
 |                  
/                   
0

\int\limits_{0}^{1} x \cos{\left(\frac{x - \pi}{3} \right)}\, dx

Integral(x*cos((x - pi)/3), (x, 0, 1))

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 $\operatorname{dv}{\left(x \right)} = \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = \frac{x}{3} + \frac{\pi}{6}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 3 \sin{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = 3 \int \sin{\left(u \right)}\, du$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- 3 \cos{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- 3 \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- 3 \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\right)\, dx = - 3 \int \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)}\, dx$
1. Let $u = \frac{x}{3} + \frac{\pi}{6}$ .
  
  Then let $du = \frac{dx}{3}$ and substitute $3 du$ :
  
  $\int 3 \cos{\left(u \right)}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = 3 \int \cos{\left(u \right)}\, du$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $3 \sin{\left(u \right)}$
  Now substitute $u$ back in:
  
  $3 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$
So, the result is: $- 9 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}$
Add the constant of integration:

$- 3 x \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}+ \mathrm{constant}$

The answer is:

- 3 x \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                                      
 |                                                       
 |      /x - pi\               /x   pi\          /x   pi\
 | x*cos|------| dx = C + 9*sin|- + --| - 3*x*cos|- + --|
 |      \  3   /               \3   6 /          \3   6 /
 |                                                       
/

\int x \cos{\left(\frac{x - \pi}{3} \right)}\, dx = C - 3 x \cos{\left(\frac{x}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{x}{3} + \frac{\pi}{6} \right)}

Simplify

The graph

Plot the graph f(x)

The answer [src]

  9        /1   pi\        /1   pi\
- - - 3*cos|- + --| + 9*sin|- + --|
  2        \3   6 /        \3   6 /

- \frac{9}{2} - 3 \cos{\left(\frac{1}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{1}{3} + \frac{\pi}{6} \right)}

  9        /1   pi\        /1   pi\
- - - 3*cos|- + --| + 9*sin|- + --|
  2        \3   6 /        \3   6 /

- \frac{9}{2} - 3 \cos{\left(\frac{1}{3} + \frac{\pi}{6} \right)} + 9 \sin{\left(\frac{1}{3} + \frac{\pi}{6} \right)}

-9/2 - 3*cos(1/3 + pi/6) + 9*sin(1/3 + pi/6)

Numerical answer [src]

0.338258415318288

0.338258415318288

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

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Integral of xcos((x-pi)/3) dx

Limits of integration:

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Piecewise:

The solution