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Integral of d{x}:
Integral of e^(x^2)
Integral of sin(x)^3
Integral of √(x^2+1)
Integral of cos(5x)
Derivative of:
x^(2*n)
Sum of series:
x^(2*n)
Limit of the function:
x^(2*n)
Identical expressions
x^(two *n)
x to the power of (2 multiply by n)
x to the power of (two multiply by n)
x(2*n)
x2*n
x^(2n)
x(2n)
x2n
x^2n
x^(2*n)dx

Integral of x^(2*n) dx

The solution

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$$\int\limits_{0}^{1} x^{2 n}\, dx$$

Integral(x^(2*n), (x, 0, 1))

Detail solution

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

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

$$\begin{cases} - \frac{0^{2 n + 1}}{2 n + 1} + \frac{1}{2 n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq - \frac{1}{2} \\\infty & \text{otherwise} \end{cases}$$

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

$$\begin{cases} - \frac{0^{2 n + 1}}{2 n + 1} + \frac{1}{2 n + 1} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq - \frac{1}{2} \\\infty & \text{otherwise} \end{cases}$$

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

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

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Identical expressions

Integral of x^(2*n) dx

Limits of integration:

The graph:

Piecewise:

The solution