Integral of y+xe^y dx

The solution

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  5              
  /              
 |               
 |  /       y\   
 |  \y + x*E / dx
 |               
/                
0

$$\int\limits_{0}^{5} \left(e^{y} x + y\right)\, dx$$

Integral(y + x*E^y, (x, 0, 5))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int e^{y} x\, dx = e^{y} \int x\, dx$
  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: $\frac{x^{2} e^{y}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int y\, dx = x y$
The result is: $\frac{x^{2} e^{y}}{2} + x y$
Now simplify:

$\frac{x \left(x e^{y} + 2 y\right)}{2}$
Add the constant of integration:

$\frac{x \left(x e^{y} + 2 y\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x \left(x e^{y} + 2 y\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                            2  y
 | /       y\                x *e 
 | \y + x*E / dx = C + x*y + -----
 |                             2  
/

$$\int \left(e^{y} x + y\right)\, dx = C + \frac{x^{2} e^{y}}{2} + x y$$

Simplify

The answer [src]

          y
      25*e 
5*y + -----
        2

$$5 y + \frac{25 e^{y}}{2}$$

          y
      25*e 
5*y + -----
        2

$$5 y + \frac{25 e^{y}}{2}$$

5*y + 25*exp(y)/2

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

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Integral of y+xe^y dx

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