Integral of arcsin dx

The solution

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

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

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

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.
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:

$- \sqrt{1 - x^{2}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

     pi
-1 + --
     2

$$-1 + \frac{\pi}{2}$$

     pi
-1 + --
     2

$$-1 + \frac{\pi}{2}$$

-1 + pi/2

Numerical answer [src]

0.570796326794897

0.570796326794897

The graph

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

Other calculators

How to use it?

Identical expressions

Integral of arcsin dx

Limits of integration:

The graph:

Piecewise:

The solution