Integral of 5x+1 dx

You have entered [src]

  b             
  /             
 |              
 |  (5*x + 1) dx
 |              
/               
a

$$\int\limits_{a}^{b} \left(5 x + 1\right)\, dx$$

Integral(5*x + 1, (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:
  
  $\int 5 x\, dx = 5 \int x\, dx$
  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: $\frac{5 x^{2}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $\frac{5 x^{2}}{2} + x$
Now simplify:

$\frac{x \left(5 x + 2\right)}{2}$
Add the constant of integration:

$\frac{x \left(5 x + 2\right)}{2}+ \mathrm{constant}$

The answer is:

$\frac{x \left(5 x + 2\right)}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          2
 |                        5*x 
 | (5*x + 1) dx = C + x + ----
 |                         2  
/

$$\int \left(5 x + 1\right)\, dx = C + \frac{5 x^{2}}{2} + x$$

Simplify

The answer [src]

           2      2
        5*a    5*b 
b - a - ---- + ----
         2      2

$$- \frac{5 a^{2}}{2} - a + \frac{5 b^{2}}{2} + b$$

           2      2
        5*a    5*b 
b - a - ---- + ----
         2      2

$$- \frac{5 a^{2}}{2} - a + \frac{5 b^{2}}{2} + b$$

b - a - 5*a^2/2 + 5*b^2/2

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