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Integral of d{x}:
Integral of 0dx
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Integral of dr/r
Integral of 1/sqrt(x^2-5)
Identical expressions
one /sqrt(x^ two - five)
1 divide by square root of (x squared minus 5)
one divide by square root of (x to the power of two minus five)
1/√(x^2-5)
1/sqrt(x2-5)
1/sqrtx2-5
1/sqrt(x²-5)
1/sqrt(x to the power of 2-5)
1/sqrtx^2-5
1 divide by sqrt(x^2-5)
1/sqrt(x^2-5)dx
Similar expressions
1/sqrt(x^2+5)

Solving integrals/ 1/sqrt(x^2-5)

Integral of 1/sqrt(x^2-5) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |       1        
 |  ----------- dx
 |     ________   
 |    /  2        
 |  \/  x  - 5    
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{1}{\sqrt{x^{2} - 5}}\, dx$$

Integral(1/(sqrt(x^2 - 5)), (x, 0, 1))

Detail solution

TrigSubstitutionRule(theta=_theta, func=sqrt(5)*sec(_theta), rewritten=sec(_theta), substep=RewriteRule(rewritten=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=tan(_theta) + sec(_theta), constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta)], context=(tan(_theta)*sec(_theta) + sec(_theta)**2)/(tan(_theta) + sec(_theta)), symbol=_theta), context=sec(_theta), symbol=_theta), restriction=(x < sqrt(5)) & (x > -sqrt(5)), context=1/(sqrt(x**2 - 5)), symbol=x)

Now simplify:

$\begin{cases} \log{\left(x + \sqrt{x^{2} - 5} \right)} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}$
Add the constant of integration:

$\begin{cases} \log{\left(x + \sqrt{x^{2} - 5} \right)} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \log{\left(x + \sqrt{x^{2} - 5} \right)} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                         
 |                      //   /                   _________\                                \
 |      1               ||   |    ___     ___   /       2 |                                |
 | ----------- dx = C + |<   |x*\/ 5    \/ 5 *\/  -5 + x  |         /       ___        ___\|
 |    ________          ||log|------- + ------------------|  for And\x > -\/ 5 , x < \/ 5 /|
 |   /  2               \\   \   5              5         /                                /
 | \/  x  - 5                                                                               
 |                                                                                          
/

$$\int \frac{1}{\sqrt{x^{2} - 5}}\, dx = C + \begin{cases} \log{\left(\frac{\sqrt{5} x}{5} + \frac{\sqrt{5} \sqrt{x^{2} - 5}}{5} \right)} & \text{for}\: x > - \sqrt{5} \wedge x < \sqrt{5} \end{cases}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

              /  ___\
  pi*I        |\/ 5 |
- ---- + acosh|-----|
   2          \  5  /

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

              /  ___\
  pi*I        |\/ 5 |
- ---- + acosh|-----|
   2          \  5  /

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

-pi*i/2 + acosh(sqrt(5)/5)

Numerical answer [src]

(0.0 - 0.463647609000806j)

(0.0 - 0.463647609000806j)

The graph

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

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Integral of 1/sqrt(x^2-5) dx

Limits of integration:

The graph:

Piecewise:

The solution