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Solving integrals/ y^x

Integral of y^x dx

Limits of integration:

from to
v

The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01yx dx\int\limits_{0}^{1} y^{x}\, dx 
Integral(y^x, (x, 0, 1))
Detail solution

    PiecewiseRule(subfunctions=[(ExpRule(base=y, exp=x, context=y**x, symbol=x), Ne(log(y), 0)), (ConstantRule(constant=1, context=1, symbol=x), True)], context=y**x, symbol=x)

  1. Add the constant of integration:

    {yxlog⁡(y)for log⁡(y)≠0xotherwise+constant\begin{cases} \frac{y^{x}}{\log{\left(y \right)}} & \text{for}\: \log{\left(y \right)} \neq 0 \\x & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer is:

{yxlog⁡(y)for log⁡(y)≠0xotherwise+constant\begin{cases} \frac{y^{x}}{\log{\left(y \right)}} & \text{for}\: \log{\left(y \right)} \neq 0 \\x & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /            //   x                   \
 |             ||  y                    |
 |  x          ||------  for log(y) != 0|
 | y  dx = C + |
∫yx dx=C+{yxlog⁡(y)for log⁡(y)≠0xotherwise\int y^{x}\, dx = C + \begin{cases} \frac{y^{x}}{\log{\left(y \right)}} & \text{for}\: \log{\left(y \right)} \neq 0 \\x & \text{otherwise} \end{cases}
Simplify
The answer [src] 
/    1        y                                      
|- ------ + ------  for Or(And(y >= 0, y < 1), y > 1)
<  log(y)   log(y)                                   
|                                                    
\        1                      otherwise
{ylog⁡(y)−1log⁡(y)for (y≥0∧y<1)∨y>11otherwise\begin{cases} \frac{y}{\log{\left(y \right)}} - \frac{1}{\log{\left(y \right)}} & \text{for}\: \left(y \geq 0 \wedge y < 1\right) \vee y > 1 \\1 & \text{otherwise} \end{cases} 
=
= 
/    1        y                                      
|- ------ + ------  for Or(And(y >= 0, y < 1), y > 1)
<  log(y)   log(y)                                   
|                                                    
\        1                      otherwise
{ylog⁡(y)−1log⁡(y)for (y≥0∧y<1)∨y>11otherwise\begin{cases} \frac{y}{\log{\left(y \right)}} - \frac{1}{\log{\left(y \right)}} & \text{for}\: \left(y \geq 0 \wedge y < 1\right) \vee y > 1 \\1 & \text{otherwise} \end{cases} 
Piecewise((-1/log(y) + y/log(y), (y > 1)∨((y >= 0)∧(y < 1))), (1, True))

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

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