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Identical expressions
(one /(x^ two - twenty-five))+(one /√(x^ two + five))
(1 divide by (x squared minus 25)) plus (1 divide by √(x squared plus 5))
(one divide by (x to the power of two minus twenty minus five)) plus (one divide by √(x to the power of two plus five))
(1/(x2-25))+(1/√(x2+5))
1/x2-25+1/√x2+5
(1/(x²-25))+(1/√(x²+5))
(1/(x to the power of 2-25))+(1/√(x to the power of 2+5))
1/x^2-25+1/√x^2+5
(1 divide by (x^2-25))+(1 divide by √(x^2+5))
(1/(x^2-25))+(1/√(x^2+5))dx
Similar expressions
(1/(x^2-25))-(1/√(x^2+5))
(1/(x^2-25))+(1/√(x^2-5))
(1/(x^2+25))+(1/√(x^2+5))

Solving integrals/ (1/(x^2-25))+(1/√(x^2+5))

Integral of (1/(x^2-25))+(1/√(x^2+5)) dx

The solution

You have entered [src]

  1                           
  /                           
 |                            
 |  /   1           1     \   
 |  |------- + -----------| dx
 |  | 2           ________|   
 |  |x  - 25     /  2     |   
 |  \          \/  x  + 5 /   
 |                            
/                             
0

$$\int\limits_{0}^{1} \left(\frac{1}{\sqrt{x^{2} + 5}} + \frac{1}{x^{2} - 25}\right)\, dx$$

Integral(1/(x^2 - 25) + 1/(sqrt(x^2 + 5)), (x, 0, 1))

Detail solution

Integrate term-by-term:
The result is: $\begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} > 25 \\- \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} < 25 \end{cases} + \log{\left(\frac{\sqrt{5} x}{5} + \sqrt{\frac{x^{2}}{5} + 1} \right)}$
Now simplify:

$\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}$
Add the constant of integration:

$\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} > 25 \\\log{\left(x + \sqrt{x^{2} + 5} \right)} - \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} - \frac{\log{\left(5 \right)}}{2} & \text{for}\: x^{2} < 25 \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                    //      /x\              \                               
                                    ||-acoth|-|              |                               
  /                                 ||      \5/        2     |      /     ________          \
 |                                  ||----------  for x  > 25|      |    /      2        ___|
 | /   1           1     \          ||    5                  |      |   /      x     x*\/ 5 |
 | |------- + -----------| dx = C + |<                       | + log|  /   1 + --  + -------|
 | | 2           ________|          ||      /x\              |      \\/        5        5   /
 | |x  - 25     /  2     |          ||-atanh|-|              |                               
 | \          \/  x  + 5 /          ||      \5/        2     |                               
 |                                  ||----------  for x  < 25|                               
/                                   \\    5                  /

$$\int \left(\frac{1}{\sqrt{x^{2} + 5}} + \frac{1}{x^{2} - 25}\right)\, dx = C + \begin{cases} - \frac{\operatorname{acoth}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} > 25 \\- \frac{\operatorname{atanh}{\left(\frac{x}{5} \right)}}{5} & \text{for}\: x^{2} < 25 \end{cases} + \log{\left(\frac{\sqrt{5} x}{5} + \sqrt{\frac{x^{2}}{5} + 1} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                         /  ___\
  log(6)   log(4)        |\/ 5 |
- ------ + ------ + asinh|-----|
    10       10          \  5  /

$$- \frac{\log{\left(6 \right)}}{10} + \frac{\log{\left(4 \right)}}{10} + \operatorname{asinh}{\left(\frac{\sqrt{5}}{5} \right)}$$

                         /  ___\
  log(6)   log(4)        |\/ 5 |
- ------ + ------ + asinh|-----|
    10       10          \  5  /

$$- \frac{\log{\left(6 \right)}}{10} + \frac{\log{\left(4 \right)}}{10} + \operatorname{asinh}{\left(\frac{\sqrt{5}}{5} \right)}$$

-log(6)/10 + log(4)/10 + asinh(sqrt(5)/5)

Numerical answer [src]

0.392960852434466

0.392960852434466

The graph

Integral of (1/(x^2-25))+(1/√(x^2+5)) dx

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

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Identical expressions

Similar expressions

Integral of (1/(x^2-25))+(1/√(x^2+5)) dx

Limits of integration:

The graph:

Piecewise:

The solution