Integral of xcos(8x)dx dx

The solution

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 |  x*cos(8*x) dx
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$$\int\limits_{0}^{1} x \cos{\left(8 x \right)}\, dx$$

Integral(x*cos(8*x), (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 $\operatorname{dv}{\left(x \right)} = \cos{\left(8 x \right)}$ .

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

To find $v{\left(x \right)}$ :
1. Let $u = 8 x$ .
  
  Then let $du = 8 dx$ and substitute $\frac{du}{8}$ :
  
  $\int \frac{\cos{\left(u \right)}}{8}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \cos{\left(u \right)}\, du = \frac{\int \cos{\left(u \right)}\, du}{8}$
    1. The integral of cosine is sine:
      
      $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
    So, the result is: $\frac{\sin{\left(u \right)}}{8}$
  Now substitute $u$ back in:
  
  $\frac{\sin{\left(8 x \right)}}{8}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\sin{\left(8 x \right)}}{8}\, dx = \frac{\int \sin{\left(8 x \right)}\, dx}{8}$
1. Let $u = 8 x$ .
  
  Then let $du = 8 dx$ and substitute $\frac{du}{8}$ :
  
  $\int \frac{\sin{\left(u \right)}}{8}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \sin{\left(u \right)}\, du = \frac{\int \sin{\left(u \right)}\, du}{8}$
    1. The integral of sine is negative cosine:
      
      $\int \sin{\left(u \right)}\, du = - \cos{\left(u \right)}$
    So, the result is: $- \frac{\cos{\left(u \right)}}{8}$
  Now substitute $u$ back in:
  
  $- \frac{\cos{\left(8 x \right)}}{8}$
So, the result is: $- \frac{\cos{\left(8 x \right)}}{64}$
Add the constant of integration:

$\frac{x \sin{\left(8 x \right)}}{8} + \frac{\cos{\left(8 x \right)}}{64}+ \mathrm{constant}$

The answer is:

$\frac{x \sin{\left(8 x \right)}}{8} + \frac{\cos{\left(8 x \right)}}{64}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                     cos(8*x)   x*sin(8*x)
 | x*cos(8*x) dx = C + -------- + ----------
 |                        64          8     
/

$$\int x \cos{\left(8 x \right)}\, dx = C + \frac{x \sin{\left(8 x \right)}}{8} + \frac{\cos{\left(8 x \right)}}{64}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1    sin(8)   cos(8)
- -- + ------ + ------
  64     8        64

$$- \frac{1}{64} + \frac{\cos{\left(8 \right)}}{64} + \frac{\sin{\left(8 \right)}}{8}$$

  1    sin(8)   cos(8)
- -- + ------ + ------
  64     8        64

$$- \frac{1}{64} + \frac{\cos{\left(8 \right)}}{64} + \frac{\sin{\left(8 \right)}}{8}$$

-1/64 + sin(8)/8 + cos(8)/64

Numerical answer [src]

0.105771342799663

0.105771342799663

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

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Identical expressions

Integral of xcos(8x)dx dx

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