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Identical expressions
(four *x^ three + one)^ five
(4 multiply by x cubed plus 1) to the power of 5
(four multiply by x to the power of three plus one) to the power of five
(4*x3+1)5
4*x3+15
(4*x³+1)⁵
(4*x to the power of 3+1) to the power of 5
(4x^3+1)^5
(4x3+1)5
4x3+15
4x^3+1^5
(4*x^3+1)^5dx
Similar expressions
(4*x^3-1)^5

Integral of (4*x^3+1)^5 dx

The solution

You have entered [src]

  1               
  /               
 |                
 |            5   
 |  /   3    \    
 |  \4*x  + 1/  dx
 |                
/                 
0

$$\int\limits_{0}^{1} \left(4 x^{3} + 1\right)^{5}\, dx$$

Integral((4*x^3 + 1)^5, (x, 0, 1))

Detail solution

Rewrite the integrand:

$\left(4 x^{3} + 1\right)^{5} = 1024 x^{15} + 1280 x^{12} + 640 x^{9} + 160 x^{6} + 20 x^{3} + 1$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 1024 x^{15}\, dx = 1024 \int x^{15}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{15}\, dx = \frac{x^{16}}{16}$
  So, the result is: $64 x^{16}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 1280 x^{12}\, dx = 1280 \int x^{12}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{12}\, dx = \frac{x^{13}}{13}$
  So, the result is: $\frac{1280 x^{13}}{13}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 640 x^{9}\, dx = 640 \int x^{9}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{9}\, dx = \frac{x^{10}}{10}$
  So, the result is: $64 x^{10}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 160 x^{6}\, dx = 160 \int x^{6}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{6}\, dx = \frac{x^{7}}{7}$
  So, the result is: $\frac{160 x^{7}}{7}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 20 x^{3}\, dx = 20 \int x^{3}\, dx$
  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: $5 x^{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $64 x^{16} + \frac{1280 x^{13}}{13} + 64 x^{10} + \frac{160 x^{7}}{7} + 5 x^{4} + x$
Now simplify:

$\frac{x \left(5824 x^{15} + 8960 x^{12} + 5824 x^{9} + 2080 x^{6} + 455 x^{3} + 91\right)}{91}$
Add the constant of integration:

$\frac{x \left(5824 x^{15} + 8960 x^{12} + 5824 x^{9} + 2080 x^{6} + 455 x^{3} + 91\right)}{91}+ \mathrm{constant}$

The answer is:

$\frac{x \left(5824 x^{15} + 8960 x^{12} + 5824 x^{9} + 2080 x^{6} + 455 x^{3} + 91\right)}{91}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                   
 |                                                                    
 |           5                                            7         13
 | /   3    \                  4       10       16   160*x    1280*x  
 | \4*x  + 1/  dx = C + x + 5*x  + 64*x   + 64*x   + ------ + --------
 |                                                     7         13   
/

$$\int \left(4 x^{3} + 1\right)^{5}\, dx = C + 64 x^{16} + \frac{1280 x^{13}}{13} + 64 x^{10} + \frac{160 x^{7}}{7} + 5 x^{4} + x$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

23234
-----
  91

$$\frac{23234}{91}$$

23234
-----
  91

$$\frac{23234}{91}$$

23234/91

Numerical answer [src]

255.318681318681

255.318681318681

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

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Integral of (4*x^3+1)^5 dx

Limits of integration:

The graph:

Piecewise:

The solution