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Solving integrals/ y^x

Integral of y^x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1      
  /      
 |       
 |   x   
 |  y  dx
 |       
/        
0
$$\int\limits_{0}^{1} y^{x}\, dx$$ 
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:

The answer is:

The answer (Indefinite) [src] 
  /            //   x                   \
 |             ||  y                    |
 |  x          ||------  for log(y) != 0|
 | y  dx = C + |
$$\int y^{x}\, dx = C + \begin{cases} \frac{y^{x}}{\log{\left(y \right)}} & \text{for}\: \log{\left(y \right)} \neq 0 \\x & \text{otherwise} \end{cases}$$
Simplify
The answer [src] 
/    1        y                                      
|- ------ + ------  for Or(And(y >= 0, y < 1), y > 1)
<  log(y)   log(y)                                   
|                                                    
\        1                      otherwise
$$\begin{cases} \frac{y}{\log{\left(y \right)}} - \frac{1}{\log{\left(y \right)}} & \text{for}\: \left(y \geq 0 \wedge y < 1\right) \vee y > 1 \\1 & \text{otherwise} \end{cases}$$ 
=
= 
/    1        y                                      
|- ------ + ------  for Or(And(y >= 0, y < 1), y > 1)
<  log(y)   log(y)                                   
|                                                    
\        1                      otherwise
$$\begin{cases} \frac{y}{\log{\left(y \right)}} - \frac{1}{\log{\left(y \right)}} & \text{for}\: \left(y \geq 0 \wedge y < 1\right) \vee y > 1 \\1 & \text{otherwise} \end{cases}$$ 
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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