Mister Exam

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Integral of d{x}:
Integral of 5^x
Integral of x^2inx
Integral of sqrt(tanhx)
Integral of dt/(5-16t^2)
Identical expressions
dt/(five -16t^ two)
dt divide by (5 minus 16t squared )
dt divide by (five minus 16t to the power of two)
dt/(5-16t2)
dt/5-16t2
dt/(5-16t²)
dt/(5-16t to the power of 2)
dt/5-16t^2
dt divide by (5-16t^2)
Similar expressions
dt/(5+16t^2)

Solving integrals/ dt/(5-16t^2)

Integral of dt/(5-16t^2) dx

The solution

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  1               
  /               
 |                
 |        1       
 |  1*--------- dt
 |            2   
 |    5 - 16*t    
 |                
/                 
0

$$\int\limits_{0}^{1} 1 \cdot \frac{1}{- 16 t^{2} + 5}\, dt$$

Integral(1/(5 - 16*t^2), (t, 0, 1))

Detail solution

Rewrite the integrand:

$1 \cdot \frac{1}{5 - 16 t^{2}} = - \frac{\sqrt{5}}{40} \left(- \frac{1}{t + \frac{\sqrt{5}}{4}} + \frac{1}{t - \frac{\sqrt{5}}{4}}\right)$
The integral of a constant times a function is the constant times the integral of the function:

$\int - \frac{\sqrt{5}}{40} \left(- \frac{1}{t + \frac{\sqrt{5}}{4}} + \frac{1}{t - \frac{\sqrt{5}}{4}}\right)\, dt = - \frac{\sqrt{5} \int \left(- \frac{1}{t + \frac{\sqrt{5}}{4}} + \frac{1}{t - \frac{\sqrt{5}}{4}}\right)\, dt}{40}$
1. Integrate term-by-term:
  1. The integral of $\frac{1}{t - \frac{\sqrt{5}}{4}}$ is $\log{\left(t - \frac{\sqrt{5}}{4} \right)}$ .
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{t + \frac{\sqrt{5}}{4}}\right)\, dt = - \int \frac{1}{t + \frac{\sqrt{5}}{4}}\, dt$
    1. The integral of $\frac{1}{t + \frac{\sqrt{5}}{4}}$ is $\log{\left(t + \frac{\sqrt{5}}{4} \right)}$ .
    So, the result is: $- \log{\left(t + \frac{\sqrt{5}}{4} \right)}$
  The result is: $\log{\left(t - \frac{\sqrt{5}}{4} \right)} - \log{\left(t + \frac{\sqrt{5}}{4} \right)}$
So, the result is: $- \frac{\sqrt{5} \left(\log{\left(t - \frac{\sqrt{5}}{4} \right)} - \log{\left(t + \frac{\sqrt{5}}{4} \right)}\right)}{40}$
Now simplify:

$\frac{\sqrt{5} \left(- \log{\left(t - \frac{\sqrt{5}}{4} \right)} + \log{\left(t + \frac{\sqrt{5}}{4} \right)}\right)}{40}$
Add the constant of integration:

$\frac{\sqrt{5} \left(- \log{\left(t - \frac{\sqrt{5}}{4} \right)} + \log{\left(t + \frac{\sqrt{5}}{4} \right)}\right)}{40}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt{5} \left(- \log{\left(t - \frac{\sqrt{5}}{4} \right)} + \log{\left(t + \frac{\sqrt{5}}{4} \right)}\right)}{40}+ \mathrm{constant}$

The answer (Indefinite) [src]

                              /     /      ___\      /      ___\\
  /                       ___ |     |    \/ 5 |      |    \/ 5 ||
 |                      \/ 5 *|- log|t + -----| + log|t - -----||
 |       1                    \     \      4  /      \      4  //
 | 1*--------- dt = C - -----------------------------------------
 |           2                              40                   
 |   5 - 16*t                                                    
 |                                                               
/

$$-{{\log \left({{32\,t-8\,\sqrt{5}}\over{32\,t+8\,\sqrt{5}}}\right) }\over{8\,\sqrt{5}}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$-{{\log \left({{11}\over{8\,\sqrt{5}+21}}\right)}\over{8\,\sqrt{5} }}$$

nan

$$\text{NaN}$$

Simplify

Numerical answer [src]

-0.220245899518467

-0.220245899518467

The graph

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

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Identical expressions

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Integral of dt/(5-16t^2) dx

Limits of integration:

The graph:

Piecewise:

The solution