Integral of (f1(x)+f2(x)) dx

The solution

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  b                 
  /                 
 |                  
 |  (f1*x + f2*x) dx
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a

$$\int\limits_{a}^{b} \left(f_{1} x + f_{2} x\right)\, dx$$

Integral(f1*x + f2*x, (x, a, b))

Detail solution

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

$\frac{x^{2} \left(f_{1} + f_{2}\right)}{2}$
Add the constant of integration:

$\frac{x^{2} \left(f_{1} + f_{2}\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \left(f_{1} + f_{2}\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                           2       2
 |                        f1*x    f2*x 
 | (f1*x + f2*x) dx = C + ----- + -----
 |                          2       2  
/

$$\int \left(f_{1} x + f_{2} x\right)\, dx = C + \frac{f_{1} x^{2}}{2} + \frac{f_{2} x^{2}}{2}$$

Simplify

The answer [src]

 2 /f1   f2\    2 /f1   f2\
b *|-- + --| - a *|-- + --|
   \2    2 /      \2    2 /

$$- a^{2} \left(\frac{f_{1}}{2} + \frac{f_{2}}{2}\right) + b^{2} \left(\frac{f_{1}}{2} + \frac{f_{2}}{2}\right)$$

 2 /f1   f2\    2 /f1   f2\
b *|-- + --| - a *|-- + --|
   \2    2 /      \2    2 /

$$- a^{2} \left(\frac{f_{1}}{2} + \frac{f_{2}}{2}\right) + b^{2} \left(\frac{f_{1}}{2} + \frac{f_{2}}{2}\right)$$

b^2*(f1/2 + f2/2) - a^2*(f1/2 + f2/2)

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

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Integral of (f1(x)+f2(x)) dx

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