Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x*x
Integral of e^(x*(-3))
Integral of sin²xcos²x
Integral of (x+2)dx
Identical expressions
2x/x*sqrt(2x+ one)/sqrt(2x)
2x divide by x multiply by square root of (2x plus 1) divide by square root of (2x)
2x divide by x multiply by square root of (2x plus one) divide by square root of (2x)
2x/x*√(2x+1)/√(2x)
2x/xsqrt(2x+1)/sqrt(2x)
2x/xsqrt2x+1/sqrt2x
2x divide by x*sqrt(2x+1) divide by sqrt(2x)
2x/x*sqrt(2x+1)/sqrt(2x)dx
Similar expressions
2x/x*sqrt(2x-1)/sqrt(2x)

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

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

The solution

You have entered [src]

  2                   
  /                   
 |                    
 |        _________   
 |  2*x*\/ 2*x + 1    
 |  --------------- dx
 |         _____      
 |     x*\/ 2*x       
 |                    
/                     
1

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

Simplify

Detail solution

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

$\int \frac{2 x \sqrt{2 x + 1}}{x \sqrt{2 x}}\, dx = 2 \int \frac{x \sqrt{2 x + 1}}{x \sqrt{2 x}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\begin{cases} \sqrt{x} \sqrt{x + \frac{1}{2}} + \frac{\operatorname{acosh}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)}}{2} & \text{for}\: \left|{x + \frac{1}{2}}\right| > \frac{1}{2} \\- \frac{i \operatorname{asin}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)}}{2} - \frac{i \left(x + \frac{1}{2}\right)^{\frac{3}{2}}}{\sqrt{- x}} + \frac{i \sqrt{x + \frac{1}{2}}}{2 \sqrt{- x}} & \text{otherwise} \end{cases}$
So, the result is: $2 \left(\begin{cases} \sqrt{x} \sqrt{x + \frac{1}{2}} + \frac{\operatorname{acosh}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)}}{2} & \text{for}\: \left|{x + \frac{1}{2}}\right| > \frac{1}{2} \\- \frac{i \operatorname{asin}{\left(\sqrt{2} \sqrt{x + \frac{1}{2}} \right)}}{2} - \frac{i \left(x + \frac{1}{2}\right)^{\frac{3}{2}}}{\sqrt{- x}} + \frac{i \sqrt{x + \frac{1}{2}}}{2 \sqrt{- x}} & \text{otherwise} \end{cases}\right)$
Now simplify:

$\begin{cases} \sqrt{2} \sqrt{x} \sqrt{2 x + 1} + \operatorname{acosh}{\left(\sqrt{2 x + 1} \right)} & \text{for}\: x > 0 \vee x < -1 \\- \frac{i x \sqrt{4 x + 2}}{\sqrt{- x}} - i \operatorname{asin}{\left(\sqrt{2 x + 1} \right)} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \sqrt{2} \sqrt{x} \sqrt{2 x + 1} + \operatorname{acosh}{\left(\sqrt{2 x + 1} \right)} & \text{for}\: x > 0 \vee x < -1 \\- \frac{i x \sqrt{4 x + 2}}{\sqrt{- x}} - i \operatorname{asin}{\left(\sqrt{2 x + 1} \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \sqrt{2} \sqrt{x} \sqrt{2 x + 1} + \operatorname{acosh}{\left(\sqrt{2 x + 1} \right)} & \text{for}\: x > 0 \vee x < -1 \\- \frac{i x \sqrt{4 x + 2}}{\sqrt{- x}} - i \operatorname{asin}{\left(\sqrt{2 x + 1} \right)} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                              //             /  ___   _________\                                                 \
  /                           ||        acosh\\/ 2 *\/ 1/2 + x /     ___   _________                             |
 |                            ||        ------------------------ + \/ x *\/ 1/2 + x           for |1/2 + x| > 1/2|
 |       _________            ||                   2                                                             |
 | 2*x*\/ 2*x + 1             ||                                                                                 |
 | --------------- dx = C + 2*|<        /  ___   _________\       _________              3/2                     |
 |        _____               ||  I*asin\\/ 2 *\/ 1/2 + x /   I*\/ 1/2 + x    I*(1/2 + x)                        |
 |    x*\/ 2*x                ||- ------------------------- + ------------- - --------------       otherwise     |
 |                            ||              2                      ____           ____                         |
/                             ||                                 2*\/ -x          \/ -x                          |
                              \\                                                                                 /

$$\sqrt{2}\,\left({{\sqrt{2\,x+1}}\over{\sqrt{x}\,\left({{2\,x+1 }\over{x}}-2\right)}}-{{\log \left({{{{2\,\sqrt{2\,x+1}}\over{\sqrt{ x}}}-2^{{{3}\over{2}}}}\over{{{2\,\sqrt{2\,x+1}}\over{\sqrt{x}}}+2^{ {{3}\over{2}}}}}\right)}\over{2^{{{3}\over{2}}}}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      /           ___      /  ___\\         /          ___      /  ___\\
  ___ |  ____   \/ 2 *acosh\\/ 5 /|     ___ |  ___   \/ 2 *acosh\\/ 3 /|
\/ 2 *|\/ 10  + ------------------| - \/ 2 *|\/ 3  + ------------------|
      \                 2         /         \                2         /

$$\sqrt{2}\,\left(-{{\log \left(9-4\,\sqrt{5}\right)}\over{2^{{{3 }\over{2}}}}}+{{\log \left(5-2^{{{3}\over{2}}}\,\sqrt{3}\right) }\over{2^{{{3}\over{2}}}}}+\sqrt{2}\,\sqrt{5}-\sqrt{3}\right)$$

      /           ___      /  ___\\         /          ___      /  ___\\
  ___ |  ____   \/ 2 *acosh\\/ 5 /|     ___ |  ___   \/ 2 *acosh\\/ 3 /|
\/ 2 *|\/ 10  + ------------------| - \/ 2 *|\/ 3  + ------------------|
      \                 2         /         \                2         /

$$- \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{3} \right)}}{2} + \sqrt{3}\right) + \sqrt{2} \left(\frac{\sqrt{2} \operatorname{acosh}{\left(\sqrt{5} \right)}}{2} + \sqrt{10}\right)$$

Simplify

Numerical answer [src]

2.32006585261462

2.32006585261462

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution