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Solving integrals/ x(4-x*cos(y)-x*sin(y))

Integral of x(4-xcos(y)-xsin(y)) dx

The solution

You have entered [src]

  2                               
  /                               
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 |  x*(4 - x*cos(y) - x*sin(y)) dx
 |                                
/                                 
0

$$\int\limits_{0}^{2} x \left(- x \sin{\left(y \right)} + \left(- x \cos{\left(y \right)} + 4\right)\right)\, dx$$

Integral(x*(4 - x*cos(y) - x*sin(y)), (x, 0, 2))

Detail solution

Rewrite the integrand:

$x \left(- x \sin{\left(y \right)} + \left(- x \cos{\left(y \right)} + 4\right)\right) = - x^{2} \sin{\left(y \right)} - x^{2} \cos{\left(y \right)} + 4 x$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x^{2} \sin{\left(y \right)}\right)\, dx = - \sin{\left(y \right)} \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $- \frac{x^{3} \sin{\left(y \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- x^{2} \cos{\left(y \right)}\right)\, dx = - \cos{\left(y \right)} \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $- \frac{x^{3} \cos{\left(y \right)}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4 x\, dx = 4 \int x\, dx$
  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: $2 x^{2}$
The result is: $- \frac{x^{3} \sin{\left(y \right)}}{3} - \frac{x^{3} \cos{\left(y \right)}}{3} + 2 x^{2}$
Now simplify:

$\frac{x^{2} \left(- \sqrt{2} x \sin{\left(y + \frac{\pi}{4} \right)} + 6\right)}{3}$
Add the constant of integration:

$\frac{x^{2} \left(- \sqrt{2} x \sin{\left(y + \frac{\pi}{4} \right)} + 6\right)}{3}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(- \sqrt{2} x \sin{\left(y + \frac{\pi}{4} \right)} + 6\right)}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                             3           3       
 |                                         2   x *cos(y)   x *sin(y)
 | x*(4 - x*cos(y) - x*sin(y)) dx = C + 2*x  - --------- - ---------
 |                                                 3           3    
/

$$\int x \left(- x \sin{\left(y \right)} + \left(- x \cos{\left(y \right)} + 4\right)\right)\, dx = C - \frac{x^{3} \sin{\left(y \right)}}{3} - \frac{x^{3} \cos{\left(y \right)}}{3} + 2 x^{2}$$

Simplify

The answer [src]

    8*cos(y)   8*sin(y)
8 - -------- - --------
       3          3

$$- \frac{8 \sin{\left(y \right)}}{3} - \frac{8 \cos{\left(y \right)}}{3} + 8$$

    8*cos(y)   8*sin(y)
8 - -------- - --------
       3          3

$$- \frac{8 \sin{\left(y \right)}}{3} - \frac{8 \cos{\left(y \right)}}{3} + 8$$

8 - 8*cos(y)/3 - 8*sin(y)/3

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

Other calculators

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Identical expressions

Similar expressions

Integral of x(4-xcos(y)-xsin(y)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of x(4-x*cos(y)-x*sin(y)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Integral of x(4-xcos(y)-xsin(y)) dx