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Solving integrals/ arcsin(x)*arcsin(x)

Integral of arcsin(x)*arcsin(x) dx

The solution

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  1                   
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 |  asin(x)*asin(x) dx
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0

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

Integral(asin(x)*asin(x), (x, 0, 1))

Detail solution

Use integration by parts:

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

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

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

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.
Use integration by parts:

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

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

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

To find $v{\left(x \right)}$ :
1. Let $u = 1 - x^{2}$ .
  
  Then let $du = - 2 x dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \left(- \frac{1}{2 \sqrt{u}}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{2}$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
    So, the result is: $- \sqrt{u}$
  Now substitute $u$ back in:
  
  $- \sqrt{1 - x^{2}}$
Now evaluate the sub-integral.
The integral of a constant is the constant times the variable of integration:

$\int \left(-2\right)\, dx = - 2 x$
Add the constant of integration:

$x \operatorname{asin}^{2}{\left(x \right)} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}+ \mathrm{constant}$

The answer is:

$x \operatorname{asin}^{2}{\left(x \right)} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                 ________        
 |                                      2          /      2         
 | asin(x)*asin(x) dx = C - 2*x + x*asin (x) + 2*\/  1 - x  *asin(x)
 |                                                                  
/

$$\int \operatorname{asin}{\left(x \right)} \operatorname{asin}{\left(x \right)}\, dx = C + x \operatorname{asin}^{2}{\left(x \right)} - 2 x + 2 \sqrt{1 - x^{2}} \operatorname{asin}{\left(x \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2
     pi 
-2 + ---
      4

$$-2 + \frac{\pi^{2}}{4}$$

       2
     pi 
-2 + ---
      4

$$-2 + \frac{\pi^{2}}{4}$$

-2 + pi^2/4

Numerical answer [src]

0.46740110027234

0.46740110027234

The graph

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

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Integral of arcsin(x)*arcsin(x) dx

Limits of integration:

The graph:

Piecewise:

The solution