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Identical expressions
(x^(y+ one))/y+ one
(x to the power of (y plus 1)) divide by y plus 1
(x to the power of (y plus one)) divide by y plus one
(x(y+1))/y+1
xy+1/y+1
x^y+1/y+1
(x^(y+1)) divide by y+1
(x^(y+1))/y+1dx
Similar expressions
(x^(y+1))/y-1
(x^(y-1))/y+1

Integral of (x^(y+1))/y+1 dx

The solution

You have entered [src]

  1                
  /                
 |                 
 |  / y + 1    \   
 |  |x         |   
 |  |------ + 1| dx
 |  \  y       /   
 |                 
/                  
0

$$\int\limits_{0}^{1} \left(\frac{x^{y + 1}}{y} + 1\right)\, dx$$

Integral(x^(y + 1)/y + 1, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x^{y + 1}}{y}\, dx = \frac{\int x^{y + 1}\, dx}{y}$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{y + 1}\, dx = \begin{cases} \frac{x^{y + 2}}{y + 2} & \text{for}\: y + 1 \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}$
  So, the result is: $\frac{\begin{cases} \frac{x^{y + 2}}{y + 2} & \text{for}\: y + 1 \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}}{y}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x + \frac{\begin{cases} \frac{x^{y + 2}}{y + 2} & \text{for}\: y + 1 \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}}{y}$
Now simplify:

$\begin{cases} \frac{x y \left(y + 2\right) + x^{y + 2}}{y \left(y + 2\right)} & \text{for}\: y \neq -2 \\x + \frac{\log{\left(x \right)}}{y} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{x y \left(y + 2\right) + x^{y + 2}}{y \left(y + 2\right)} & \text{for}\: y \neq -2 \\x + \frac{\log{\left(x \right)}}{y} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{x y \left(y + 2\right) + x^{y + 2}}{y \left(y + 2\right)} & \text{for}\: y \neq -2 \\x + \frac{\log{\left(x \right)}}{y} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                             / 2 + y                 
                             |x                      
                             |------  for y + 1 != -1
  /                          <2 + y                  
 |                           |                       
 | / y + 1    \              |log(x)     otherwise   
 | |x         |              \                       
 | |------ + 1| dx = C + x + ------------------------
 | \  y       /                         y            
 |                                                   
/

$$\int \left(\frac{x^{y + 1}}{y} + 1\right)\, dx = C + x + \frac{\begin{cases} \frac{x^{y + 2}}{y + 2} & \text{for}\: y + 1 \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}}{y}$$

Simplify

The answer [src]

    //               2 + y                                    \
    ||    1         0                                         |
    ||--------- - ---------  for And(y > -oo, y < oo, y != -2)|
    ||y*(2 + y)   y*(2 + y)                                   |
1 + |<                                                        |
    ||            /1\                                         |
    ||     oo*sign|-|                    otherwise            |
    ||            \y/                                         |
    \\                                                        /

$$\begin{cases} - \frac{0^{y + 2}}{y \left(y + 2\right)} + \frac{1}{y \left(y + 2\right)} & \text{for}\: y > -\infty \wedge y < \infty \wedge y \neq -2 \\\infty \operatorname{sign}{\left(\frac{1}{y} \right)} & \text{otherwise} \end{cases} + 1$$

    //               2 + y                                    \
    ||    1         0                                         |
    ||--------- - ---------  for And(y > -oo, y < oo, y != -2)|
    ||y*(2 + y)   y*(2 + y)                                   |
1 + |<                                                        |
    ||            /1\                                         |
    ||     oo*sign|-|                    otherwise            |
    ||            \y/                                         |
    \\                                                        /

1 + Piecewise((1/(y*(2 + y)) - 0^(2 + y)/(y*(2 + y)), (y > -oo)∧(y < oo)∧(Ne(y, -2))), (oo*sign(1/y), True))

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

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Similar expressions

Integral of (x^(y+1))/y+1 dx

Limits of integration:

The graph:

Piecewise:

The solution