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(x-sinx)/x^(1/3)

Integral of (x+sinx)/x^(1/3) dx

The solution

You have entered [src]

 oo              
  /              
 |               
 |  x + sin(x)   
 |  ---------- dx
 |    3 ___      
 |    \/ x       
 |               
/                
1

$$\int\limits_{1}^{\infty} \frac{x + \sin{\left(x \right)}}{\sqrt[3]{x}}\, dx$$

Integral((x + sin(x))/x^(1/3), (x, 1, oo))

Detail solution

Rewrite the integrand:

$\frac{x + \sin{\left(x \right)}}{\sqrt[3]{x}} = \frac{x}{\sqrt[3]{x}} + \frac{\sin{\left(x \right)}}{\sqrt[3]{x}}$
Integrate term-by-term:
1. Let $u = \frac{1}{\sqrt[3]{x}}$ .
  
  Then let $du = - \frac{dx}{3 x^{\frac{4}{3}}}$ and substitute $- 3 du$ :
  
  $\int \left(- \frac{3}{u^{6}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u^{6}}\, du = - 3 \int \frac{1}{u^{6}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
    So, the result is: $\frac{3}{5 u^{5}}$
  Now substitute $u$ back in:
  
  $\frac{3 x^{\frac{5}{3}}}{5}$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x^{\frac{5}{3}} \Gamma\left(\frac{5}{6}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)}}{2 \Gamma\left(\frac{11}{6}\right)}$
The result is: $\frac{x^{\frac{5}{3}} \Gamma\left(\frac{5}{6}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)}}{2 \Gamma\left(\frac{11}{6}\right)} + \frac{3 x^{\frac{5}{3}}}{5}$
Now simplify:

$\frac{3 x^{\frac{5}{3}} \left({{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)} + 1\right)}{5}$
Add the constant of integration:

$\frac{3 x^{\frac{5}{3}} \left({{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)} + 1\right)}{5}+ \mathrm{constant}$

The answer is:

$\frac{3 x^{\frac{5}{3}} \left({{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)} + 1\right)}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                                       
                                                  _  /          |   2 \
  /                              5/3             |_  |   5/6    | -x  |
 |                        5/3   x   *Gamma(5/6)* |   |          | ----|
 | x + sin(x)          3*x                      1  2 \3/2, 11/6 |  4  /
 | ---------- dx = C + ------ + ---------------------------------------
 |   3 ___               5                   2*Gamma(11/6)             
 |   \/ x                                                              
 |                                                                     
/

$$\int \frac{x + \sin{\left(x \right)}}{\sqrt[3]{x}}\, dx = C + \frac{x^{\frac{5}{3}} \Gamma\left(\frac{5}{6}\right) {{}_{1}F_{2}\left(\begin{matrix} \frac{5}{6} \\ \frac{3}{2}, \frac{11}{6} \end{matrix}\middle| {- \frac{x^{2}}{4}} \right)}}{2 \Gamma\left(\frac{11}{6}\right)} + \frac{3 x^{\frac{5}{3}}}{5}$$

Simplify

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

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Integral of (x+sinx)/x^(1/3) dx

Limits of integration:

The graph:

Piecewise:

The solution