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Solving integrals/ 2*x^(n-1)/factorial(n-1)

Integral of 2*x^(n-1)/factorial(n-1) dx

The solution

You have entered [src]

  1            
  /            
 |             
 |     n - 1   
 |  2*x        
 |  -------- dx
 |  (n - 1)!   
 |             
/              
0

$$\int\limits_{0}^{1} \frac{2 x^{n - 1}}{\left(n - 1\right)!}\, dx$$

Integral((2*x^(n - 1))/factorial(n - 1), (x, 0, 1))

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

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

$\begin{cases} \frac{2 x^{n}}{\Gamma\left(n + 1\right)} & \text{for}\: n \neq 0 \\\frac{2 \log{\left(x \right)}}{\Gamma\left(n\right)} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{2 x^{n}}{\Gamma\left(n + 1\right)} & \text{for}\: n \neq 0 \\\frac{2 \log{\left(x \right)}}{\Gamma\left(n\right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{2 x^{n}}{\Gamma\left(n + 1\right)} & \text{for}\: n \neq 0 \\\frac{2 \log{\left(x \right)}}{\Gamma\left(n\right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

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

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

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

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

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

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Integral of 2*x^(n-1)/factorial(n-1) dx

Limits of integration:

The graph:

Piecewise:

The solution