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Similar expressions
x*(sin(x-y))

Integral of x*(sin(x+y)) dx

The solution

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  p                
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 |  x*sin(x + y) dx
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0

$$\int\limits_{0}^{p} x \sin{\left(x + y \right)}\, dx$$

Integral(x*sin(x + y), (x, 0, p))

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 $\operatorname{dv}{\left(x \right)} = \sin{\left(x + y \right)}$ .

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

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

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

$- x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}+ \mathrm{constant}$

The answer is:

$- x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
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 | x*sin(x + y) dx = C - x*cos(x + y) + sin(x + y)
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$$\int x \sin{\left(x + y \right)}\, dx = C - x \cos{\left(x + y \right)} + \sin{\left(x + y \right)}$$

Simplify

The answer [src]

-sin(y) - p*cos(p + y) + sin(p + y)

$$- p \cos{\left(p + y \right)} - \sin{\left(y \right)} + \sin{\left(p + y \right)}$$

-sin(y) - p*cos(p + y) + sin(p + y)

$$- p \cos{\left(p + y \right)} - \sin{\left(y \right)} + \sin{\left(p + y \right)}$$

-sin(y) - p*cos(p + y) + sin(p + y)

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

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Identical expressions

Similar expressions

Integral of x*(sin(x+y)) dx

Limits of integration:

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Piecewise:

The solution