Mister Exam

Other calculators

Solving integrals/ ax^n

Integral of ax^n dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1        
  /        
 |         
 |     n   
 |  a*x  dx
 |         
/          
0
$$\int\limits_{0}^{1} a x^{n}\, dx$$ 
Integral(a*x^n, (x, 0, 1))
Detail solution

  1. The integral of a constant times a function is the constant times the integral of the function:

    1. The integral of is when :

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:

The answer is:

The answer (Indefinite) [src] 
  /                // 1 + n             \
 |                 ||x                  |
 |    n            ||------  for n != -1|
 | a*x  dx = C + a*|<1 + n              |
 |                 ||                   |
/                  ||log(x)   otherwise |
                   \\                   /
$$\int a x^{n}\, dx = C + a \left(\begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
Simplify
The answer [src] 
/           1 + n                                   
|  a     a*0                                        
|----- - --------  for And(n > -oo, n < oo, n != -1)
<1 + n    1 + n                                     
|                                                   
|   oo*sign(a)                 otherwise            
\
$$\begin{cases} - \frac{0^{n + 1} a}{n + 1} + \frac{a}{n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq -1 \\\infty \operatorname{sign}{\left(a \right)} & \text{otherwise} \end{cases}$$ 
=
= 
/           1 + n                                   
|  a     a*0                                        
|----- - --------  for And(n > -oo, n < oo, n != -1)
<1 + n    1 + n                                     
|                                                   
|   oo*sign(a)                 otherwise            
\
$$\begin{cases} - \frac{0^{n + 1} a}{n + 1} + \frac{a}{n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq -1 \\\infty \operatorname{sign}{\left(a \right)} & \text{otherwise} \end{cases}$$ 
Piecewise((a/(1 + n) - a*0^(1 + n)/(1 + n), (n > -oo)∧(n < oo)∧(Ne(n, -1))), (oo*sign(a), True))

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

    © Mister Exam – Calculator