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Solving integrals/ nx^(n-1)

Integral of nx^(n-1) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1            
  /            
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 |     n - 1   
 |  n*x      dx
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0
∫01nxn−1 dx\int\limits_{0}^{1} n x^{n - 1}\, dx 
Integral(n*x^(n - 1), (x, 0, 1))
Detail solution

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

    ∫nxn−1 dx=n∫xn−1 dx\int n x^{n - 1}\, dx = n \int x^{n - 1}\, dx

    1. The integral of xnx^{n} is xn+1n+1\frac{x^{n + 1}}{n + 1} when n≠−1n \neq -1:

      ∫xn−1 dx={xnnfor n−1≠−1log⁡(x)otherwise\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: n({xnnfor n−1≠−1log⁡(x)otherwise)n \left(\begin{cases} \frac{x^{n}}{n} & \text{for}\: n - 1 \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}\right)

  2. Now simplify:

    {xnfor n≠0nlog⁡(x)otherwise\begin{cases} x^{n} & \text{for}\: n \neq 0 \\n \log{\left(x \right)} & \text{otherwise} \end{cases}

  3. Add the constant of integration:

    {xnfor n≠0nlog⁡(x)otherwise+constant\begin{cases} x^{n} & \text{for}\: n \neq 0 \\n \log{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}

The answer is:

{xnfor n≠0nlog⁡(x)otherwise+constant\begin{cases} x^{n} & \text{for}\: n \neq 0 \\n \log{\left(x \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}

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

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

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