Integral of (x^k)dx dx

You have entered [src]

$$\int\limits_{0}^{\infty} x^{k}\, dx$$

Integral(x^k, (x, 0, oo))

Detail solution

The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :

$\int x^{k}\, dx = \begin{cases} \frac{x^{k + 1}}{k + 1} & \text{for}\: k \neq -1 \\\log{\left(x \right)} & \text{otherwise} \end{cases}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

/    0      for And(re(k) > -1, re(k) < -1)
|                                          
| oo                                       
|  /                                       
| |                                        
< |   k                                    
| |  x  dx             otherwise           
| |                                        
|/                                         
|0                                         
\

$$\begin{cases} 0 & \text{for}\: \operatorname{re}{\left(k\right)} > -1 \wedge \operatorname{re}{\left(k\right)} < -1 \\\int\limits_{0}^{\infty} x^{k}\, dx & \text{otherwise} \end{cases}$$

=

/    0      for And(re(k) > -1, re(k) < -1)
|                                          
| oo                                       
|  /                                       
| |                                        
< |   k                                    
| |  x  dx             otherwise           
| |                                        
|/                                         
|0                                         
\

$$\begin{cases} 0 & \text{for}\: \operatorname{re}{\left(k\right)} > -1 \wedge \operatorname{re}{\left(k\right)} < -1 \\\int\limits_{0}^{\infty} x^{k}\, dx & \text{otherwise} \end{cases}$$

Piecewise((0, (re(k) > -1)∧(re(k) < -1)), (Integral(x^k, (x, 0, oo)), True))

Other calculators

How to use it?

Identical expressions

Integral of (x^k)dx dx

Limits of integration:

The graph:

Piecewise:

The solution