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Solving integrals/ dx/(sqrt(2)*x+1)

Integral of dx/(sqrt(2)*x+1) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  0               
  /               
 |                
 |       1        
 |  ----------- dx
 |    ___         
 |  \/ 2 *x + 1   
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4
∫4012x+1 dx\int\limits_{4}^{0} \frac{1}{\sqrt{2} x + 1}\, dx 
Integral(1/(sqrt(2)*x + 1), (x, 4, 0))
Detail solution

  1. Let u=2x+1u = \sqrt{2} x + 1.

    Then let du=2dxdu = \sqrt{2} dx and substitute 2du2\frac{\sqrt{2} du}{2}:

    ∫22u du\int \frac{\sqrt{2}}{2 u}\, du

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

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

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

      So, the result is: 2log⁡(u)2\frac{\sqrt{2} \log{\left(u \right)}}{2}

    Now substitute uu back in:

    2log⁡(2x+1)2\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}

  2. Now simplify:

    2log⁡(2x+1)2\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}

  3. Add the constant of integration:

    2log⁡(2x+1)2+constant\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}+ \mathrm{constant}

The answer is:

2log⁡(2x+1)2+constant\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                           
 |                        ___    /  ___      \
 |      1               \/ 2 *log\\/ 2 *x + 1/
 | ----------- dx = C + ----------------------
 |   ___                          2           
 | \/ 2 *x + 1                                
 |                                            
/
∫12x+1 dx=C+2log⁡(2x+1)2\int \frac{1}{\sqrt{2} x + 1}\, dx = C + \frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}
Simplify
The graph
0.04.00.51.01.52.02.53.03.502
Plot the graph f(x)
The answer [src] 
   ___    /        ___\ 
-\/ 2 *log\1 + 4*\/ 2 / 
------------------------
           2
−2log⁡(1+42)2- \frac{\sqrt{2} \log{\left(1 + 4 \sqrt{2} \right)}}{2} 
=
= 
   ___    /        ___\ 
-\/ 2 *log\1 + 4*\/ 2 / 
------------------------
           2
−2log⁡(1+42)2- \frac{\sqrt{2} \log{\left(1 + 4 \sqrt{2} \right)}}{2} 
-sqrt(2)*log(1 + 4*sqrt(2))/2
Numerical answer [src] 
-1.34042487534557
-1.34042487534557

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

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