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Solving integrals/ x^n

Integral of x^n dx

Limits of integration:

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Piecewise:

The solution

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  1      
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∫01xn dx\int\limits_{0}^{1} x^{n}\, dx 
Integral(x^n, (x, 0, 1))
Detail solution

  1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

    ∫xn dx={xn+1n+1for n≠−1log⁡(x)otherwise\int x^{n}\, dx = \begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}

  2. Add the constant of integration:

    {xn+1n+1for n≠−1log⁡(x)otherwise+constant\begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer is:

{xn+1n+1for n≠−1log⁡(x)otherwise+constant\begin{cases} \frac{x^{n + 1}}{n + 1} & \text{for}\: n \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}

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

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

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