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Solving integrals/ x^y

Integral of x^y dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1      
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∫01xy dx\int\limits_{0}^{1} x^{y}\, dx 
Integral(x^y, (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:

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

  2. Add the constant of integration:

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

The answer is:

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

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

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

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