Integral of cbrt(x+1) dx

The solution

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  5             
  /             
 |              
 |  3 _______   
 |  \/ x + 1  dx
 |              
/               
1

$$\int\limits_{1}^{5} \sqrt[3]{x + 1}\, dx$$

Integral((x + 1)^(1/3), (x, 1, 5))

Detail solution

Let $u = x + 1$ .

Then let $du = dx$ and substitute $du$ :

$\int \sqrt[3]{u}\, du$
1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int \sqrt[3]{u}\, du = \frac{3 u^{\frac{4}{3}}}{4}$
Now substitute $u$ back in:

$\frac{3 \left(x + 1\right)^{\frac{4}{3}}}{4}$
Now simplify:

$\frac{3 \left(x + 1\right)^{\frac{4}{3}}}{4}$
Add the constant of integration:

$\frac{3 \left(x + 1\right)^{\frac{4}{3}}}{4}+ \mathrm{constant}$

The answer is:

$\frac{3 \left(x + 1\right)^{\frac{4}{3}}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                             4/3
 | 3 _______          3*(x + 1)   
 | \/ x + 1  dx = C + ------------
 |                         4      
/

$$\int \sqrt[3]{x + 1}\, dx = C + \frac{3 \left(x + 1\right)^{\frac{4}{3}}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

    3 ___     3 ___
  3*\/ 2    9*\/ 6 
- ------- + -------
     2         2

$$- \frac{3 \sqrt[3]{2}}{2} + \frac{9 \sqrt[3]{6}}{2}$$

    3 ___     3 ___
  3*\/ 2    9*\/ 6 
- ------- + -------
     2         2

$$- \frac{3 \sqrt[3]{2}}{2} + \frac{9 \sqrt[3]{6}}{2}$$

-3*2^(1/3)/2 + 9*6^(1/3)/2

Numerical answer [src]

6.28716109290232

6.28716109290232

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

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Integral of cbrt(x+1) dx

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The solution