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Solving integrals/ 1/((2-x)^(2/3))

Integral of 1/((2-x)^(2/3)) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  6              
  /              
 |               
 |      1        
 |  ---------- dx
 |         2/3   
 |  (2 - x)      
 |               
/                
2
∫261(2−x)23 dx\int\limits_{2}^{6} \frac{1}{\left(2 - x\right)^{\frac{2}{3}}}\, dx 
Integral(1/((2 - x)^(2/3)), (x, 2, 6))
Detail solution

  1. Let u=(2−x)23u = \left(2 - x\right)^{\frac{2}{3}}.

    Then let du=−2dx32−x3du = - \frac{2 dx}{3 \sqrt[3]{2 - x}} and substitute −3du2- \frac{3 du}{2}:

    ∫(−32u) du\int \left(- \frac{3}{2 \sqrt{u}}\right)\, du

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫1u du=−3∫1u du2\int \frac{1}{\sqrt{u}}\, du = - \frac{3 \int \frac{1}{\sqrt{u}}\, du}{2}

      1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

        ∫1u du=2u\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}

      So, the result is: −3u- 3 \sqrt{u}

    Now substitute uu back in:

    −32−x3- 3 \sqrt[3]{2 - x}

  2. Add the constant of integration:

    −32−x3+constant- 3 \sqrt[3]{2 - x}+ \mathrm{constant}

The answer is:

−32−x3+constant- 3 \sqrt[3]{2 - x}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                                
 |     1                 3 _______
 | ---------- dx = C - 3*\/ 2 - x 
 |        2/3                     
 | (2 - x)                        
 |                                
/
∫1(2−x)23 dx=C−32−x3\int \frac{1}{\left(2 - x\right)^{\frac{2}{3}}}\, dx = C - 3 \sqrt[3]{2 - x}
Simplify
The graph
2.000002.000102.000202.000302.000402.000502.000602.000702.000802.000902.0010001
Plot the graph f(x)
The answer [src] 
   3 ____  2/3
-3*\/ -1 *2
−3−13⋅223- 3 \sqrt[3]{-1} \cdot 2^{\frac{2}{3}} 
=
= 
   3 ____  2/3
-3*\/ -1 *2
−3−13⋅223- 3 \sqrt[3]{-1} \cdot 2^{\frac{2}{3}} 
-3*(-1)^(1/3)*2^(2/3)
Numerical answer [src] 
(-2.38110059391664 - 4.12418720659605j)
(-2.38110059391664 - 4.12418720659605j)

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

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