Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of sqrt(x)/x
Integral of (-x)/(1+x^2)
Integral of (x^2-1)e^x
Integral of ln(x)*ln(x)
Identical expressions
dx/(sqrt(two)*x+ one)
dx divide by ( square root of (2) multiply by x plus 1)
dx divide by ( square root of (two) multiply by x plus one)
dx/(√(2)*x+1)
dx/(sqrt(2)x+1)
dx/sqrt2x+1
dx divide by (sqrt(2)*x+1)
Similar expressions
dx/(sqrt(2)*x-1)

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

The solution

You have entered [src]

  0               
  /               
 |                
 |       1        
 |  ----------- dx
 |    ___         
 |  \/ 2 *x + 1   
 |                
/                 
4

$$\int\limits_{4}^{0} \frac{1}{\sqrt{2} x + 1}\, dx$$

Integral(1/(sqrt(2)*x + 1), (x, 4, 0))

Detail solution

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

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

$\int \frac{\sqrt{2}}{2 u}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u}\, du = \frac{\sqrt{2} \int \frac{1}{u}\, du}{2}$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  So, the result is: $\frac{\sqrt{2} \log{\left(u \right)}}{2}$
Now substitute $u$ back in:

$\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}$
Now simplify:

$\frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                           
 |                        ___    /  ___      \
 |      1               \/ 2 *log\\/ 2 *x + 1/
 | ----------- dx = C + ----------------------
 |   ___                          2           
 | \/ 2 *x + 1                                
 |                                            
/

$$\int \frac{1}{\sqrt{2} x + 1}\, dx = C + \frac{\sqrt{2} \log{\left(\sqrt{2} x + 1 \right)}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   ___    /        ___\ 
-\/ 2 *log\1 + 4*\/ 2 / 
------------------------
           2

$$- \frac{\sqrt{2} \log{\left(1 + 4 \sqrt{2} \right)}}{2}$$

   ___    /        ___\ 
-\/ 2 *log\1 + 4*\/ 2 / 
------------------------
           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.

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution