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Identical expressions
arcsinsqrt(one -5x^ two)
arc sinus of square root of (1 minus 5x squared )
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arcsin√(1-5x^2)
arcsinsqrt(1-5x2)
arcsinsqrt1-5x2
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arcsinsqrt(1-5x to the power of 2)
arcsinsqrt1-5x^2
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Similar expressions
arcsinsqrt(1+5x^2)

Integral of arcsinsqrt(1-5x^2) dx

The solution

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  1                       
  /                       
 |                        
 |      /   __________\   
 |      |  /        2 |   
 |  asin\\/  1 - 5*x  / dx
 |                        
/                         
0

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

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

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \operatorname{asin}{\left(\sqrt{1 - 5 x^{2}} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{\sqrt{5} x}{\sqrt{1 - 5 x^{2}} \sqrt{x^{2}}}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\sqrt{5} x^{2}}{\sqrt{1 - 5 x^{2}} \sqrt{x^{2}}}\right)\, dx = - \sqrt{5} \int \frac{x^{2}}{\sqrt{1 - 5 x^{2}} \sqrt{x^{2}}}\, dx$
So, the result is: $- \sqrt{5} \left(\begin{cases} - \frac{\sqrt{1 - 5 x^{2}}}{5} & \text{for}\: x > - \frac{\sqrt{5}}{5} \wedge x < \frac{\sqrt{5}}{5} \end{cases}\right)$
Now simplify:

$\begin{cases} x \operatorname{asin}{\left(\sqrt{1 - 5 x^{2}} \right)} - \frac{\sqrt{5 - 25 x^{2}}}{5} & \text{for}\: x > - \frac{\sqrt{5}}{5} \wedge x < \frac{\sqrt{5}}{5} \end{cases}$
Add the constant of integration:

$\begin{cases} x \operatorname{asin}{\left(\sqrt{1 - 5 x^{2}} \right)} - \frac{\sqrt{5 - 25 x^{2}}}{5} & \text{for}\: x > - \frac{\sqrt{5}}{5} \wedge x < \frac{\sqrt{5}}{5} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} x \operatorname{asin}{\left(\sqrt{1 - 5 x^{2}} \right)} - \frac{\sqrt{5 - 25 x^{2}}}{5} & \text{for}\: x > - \frac{\sqrt{5}}{5} \wedge x < \frac{\sqrt{5}}{5} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                              
 |                                                                                                               
 |     /   __________\                /   __________\         //    __________                                  \
 |     |  /        2 |                |  /        2 |     ___ ||   /        2           /       ___         ___\|
 | asin\\/  1 - 5*x  / dx = C + x*asin\\/  1 - 5*x  / + \/ 5 *|<-\/  1 - 5*x            |    -\/ 5        \/ 5 ||
 |                                                            ||---------------  for And|x > -------, x < -----||
/                                                             \\       5                \       5           5  //

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___                      ___
\/ 5                 2*I*\/ 5 
----- + I*asinh(2) - ---------
  5                      5

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

  ___                      ___
\/ 5                 2*I*\/ 5 
----- + I*asinh(2) - ---------
  5                      5

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

sqrt(5)/5 + i*asinh(2) - 2*i*sqrt(5)/5

Numerical answer [src]

(0.447174177008474 + 0.549472714110548j)

(0.447174177008474 + 0.549472714110548j)

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

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Integral of arcsinsqrt(1-5x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution