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Identical expressions
x(one +x)^ five
x(1 plus x) to the power of 5
x(one plus x) to the power of five
x(1+x)5
x1+x5
x(1+x)⁵
x1+x^5
x(1+x)^5dx
Similar expressions
x(1-x)^5

Solving integrals/ x(1+x)^5

Integral of x(1+x)^5 dx

The solution

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  1              
  /              
 |               
 |           5   
 |  x*(1 + x)  dx
 |               
/                
0

$$\int\limits_{0}^{1} x \left(x + 1\right)^{5}\, dx$$

Integral(x*(1 + x)^5, (x, 0, 1))

Detail solution

Rewrite the integrand:

$x \left(x + 1\right)^{5} = x^{6} + 5 x^{5} + 10 x^{4} + 10 x^{3} + 5 x^{2} + x$
Integrate term-by-term:
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x^{6}\, dx = \frac{x^{7}}{7}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5 x^{5}\, dx = 5 \int x^{5}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{5}\, dx = \frac{x^{6}}{6}$
  So, the result is: $\frac{5 x^{6}}{6}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 10 x^{4}\, dx = 10 \int x^{4}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{4}\, dx = \frac{x^{5}}{5}$
  So, the result is: $2 x^{5}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 10 x^{3}\, dx = 10 \int x^{3}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{3}\, dx = \frac{x^{4}}{4}$
  So, the result is: $\frac{5 x^{4}}{2}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5 x^{2}\, dx = 5 \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $\frac{5 x^{3}}{3}$
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
The result is: $\frac{x^{7}}{7} + \frac{5 x^{6}}{6} + 2 x^{5} + \frac{5 x^{4}}{2} + \frac{5 x^{3}}{3} + \frac{x^{2}}{2}$
Now simplify:

$\frac{x^{2} \cdot \left(6 x^{5} + 35 x^{4} + 84 x^{3} + 105 x^{2} + 70 x + 21\right)}{42}$
Add the constant of integration:

$\frac{x^{2} \cdot \left(6 x^{5} + 35 x^{4} + 84 x^{3} + 105 x^{2} + 70 x + 21\right)}{42}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \cdot \left(6 x^{5} + 35 x^{4} + 84 x^{3} + 105 x^{2} + 70 x + 21\right)}{42}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                       
 |                      2           7      4      3      6
 |          5          x       5   x    5*x    5*x    5*x 
 | x*(1 + x)  dx = C + -- + 2*x  + -- + ---- + ---- + ----
 |                     2           7     2      3      6  
/

$${{6\,x^7+35\,x^6+84\,x^5+105\,x^4+70\,x^3+21\,x^2}\over{42}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

107
---
 14

$${{107}\over{14}}$$

107
---
 14

$$\frac{107}{14}$$

Упростить

Numerical answer [src]

7.64285714285714

7.64285714285714

The graph

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

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Similar expressions

Integral of x(1+x)^5 dx

Limits of integration:

The graph:

Piecewise:

The solution