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Solving integrals/ dx/(sqrt(5)(x)-1)

Integral of dx/(sqrt(5)(x)-1) dx

Limits of integration:

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The graph:

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The solution

You have entered [src] 
  2               
  /               
 |                
 |       1        
 |  ----------- dx
 |    ___         
 |  \/ 5 *x - 1   
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1
∫1215x−1 dx\int\limits_{1}^{2} \frac{1}{\sqrt{5} x - 1}\, dx 
Integral(1/(sqrt(5)*x - 1), (x, 1, 2))
Detail solution

  1. Let u=5x−1u = \sqrt{5} x - 1.

    Then let du=5dxdu = \sqrt{5} dx and substitute 5du5\frac{\sqrt{5} du}{5}:

    ∫55u du\int \frac{\sqrt{5}}{5 u}\, du

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

      ∫1u du=5∫1u du5\int \frac{1}{u}\, du = \frac{\sqrt{5} \int \frac{1}{u}\, du}{5}

      1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

      So, the result is: 5log⁡(u)5\frac{\sqrt{5} \log{\left(u \right)}}{5}

    Now substitute uu back in:

    5log⁡(5x−1)5\frac{\sqrt{5} \log{\left(\sqrt{5} x - 1 \right)}}{5}

  2. Now simplify:

    5log⁡(5x−1)5\frac{\sqrt{5} \log{\left(\sqrt{5} x - 1 \right)}}{5}

  3. Add the constant of integration:

    5log⁡(5x−1)5+constant\frac{\sqrt{5} \log{\left(\sqrt{5} x - 1 \right)}}{5}+ \mathrm{constant}

The answer is:

5log⁡(5x−1)5+constant\frac{\sqrt{5} \log{\left(\sqrt{5} x - 1 \right)}}{5}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                           
 |                        ___    /  ___      \
 |      1               \/ 5 *log\\/ 5 *x - 1/
 | ----------- dx = C + ----------------------
 |   ___                          5           
 | \/ 5 *x - 1                                
 |                                            
/
∫15x−1 dx=C+5log⁡(5x−1)5\int \frac{1}{\sqrt{5} x - 1}\, dx = C + \frac{\sqrt{5} \log{\left(\sqrt{5} x - 1 \right)}}{5}
Simplify
The graph
1.002.001.101.201.301.401.501.601.701.801.900.01.0
Plot the graph f(x)
The answer [src] 
    ___    /       ___\     ___    /         ___\
  \/ 5 *log\-1 + \/ 5 /   \/ 5 *log\-1 + 2*\/ 5 /
- --------------------- + -----------------------
            5                        5
−5log⁡(−1+5)5+5log⁡(−1+25)5- \frac{\sqrt{5} \log{\left(-1 + \sqrt{5} \right)}}{5} + \frac{\sqrt{5} \log{\left(-1 + 2 \sqrt{5} \right)}}{5} 
=
= 
    ___    /       ___\     ___    /         ___\
  \/ 5 *log\-1 + \/ 5 /   \/ 5 *log\-1 + 2*\/ 5 /
- --------------------- + -----------------------
            5                        5
−5log⁡(−1+5)5+5log⁡(−1+25)5- \frac{\sqrt{5} \log{\left(-1 + \sqrt{5} \right)}}{5} + \frac{\sqrt{5} \log{\left(-1 + 2 \sqrt{5} \right)}}{5} 
-sqrt(5)*log(-1 + sqrt(5))/5 + sqrt(5)*log(-1 + 2*sqrt(5))/5
Numerical answer [src] 
0.461897674162808
0.461897674162808

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

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