Integral of xarcsinax dx

The solution

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 |  x*asin(a*x) dx
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0

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

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

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

To find $v{\left(x \right)}$ :
1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
  
  $\int x\, dx = \frac{x^{2}}{2}$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{a x^{2}}{2 \sqrt{- a^{2} x^{2} + 1}}\, dx = \frac{a \int \frac{x^{2}}{\sqrt{- a^{2} x^{2} + 1}}\, dx}{2}$
So, the result is: $\frac{a \left(\begin{cases} - \frac{x \sqrt{- a^{2} x^{2} + 1}}{2 a^{2}} + \frac{\log{\left(- 2 a^{2} x + 2 \sqrt{- a^{2}} \sqrt{- a^{2} x^{2} + 1} \right)}}{2 a^{2} \sqrt{- a^{2}}} & \text{for}\: a^{2} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}\right)}{2}$
Now simplify:

$\begin{cases} \frac{2 x^{2} \operatorname{asin}{\left(a x \right)} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{a} - \frac{\log{\left(- a^{2} x + \sqrt{- a^{2}} \sqrt{- a^{2} x^{2} + 1} \right)}}{a \sqrt{- a^{2}}} - \frac{\log{\left(2 \right)}}{a \sqrt{- a^{2}}}}{4} & \text{for}\: a > 0 \vee a < 0 \\\frac{x^{2} \left(- a x + 3 \operatorname{asin}{\left(a x \right)}\right)}{6} & \text{otherwise} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{2 x^{2} \operatorname{asin}{\left(a x \right)} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{a} - \frac{\log{\left(- a^{2} x + \sqrt{- a^{2}} \sqrt{- a^{2} x^{2} + 1} \right)}}{a \sqrt{- a^{2}}} - \frac{\log{\left(2 \right)}}{a \sqrt{- a^{2}}}}{4} & \text{for}\: a > 0 \vee a < 0 \\\frac{x^{2} \left(- a x + 3 \operatorname{asin}{\left(a x \right)}\right)}{6} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{2 x^{2} \operatorname{asin}{\left(a x \right)} + \frac{x \sqrt{- a^{2} x^{2} + 1}}{a} - \frac{\log{\left(- a^{2} x + \sqrt{- a^{2}} \sqrt{- a^{2} x^{2} + 1} \right)}}{a \sqrt{- a^{2}}} - \frac{\log{\left(2 \right)}}{a \sqrt{- a^{2}}}}{4} & \text{for}\: a > 0 \vee a < 0 \\\frac{x^{2} \left(- a x + 3 \operatorname{asin}{\left(a x \right)}\right)}{6} & \text{otherwise} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                         //   /                _____    ___________\        ___________             \
                                         ||   |       2       /   2    /      2  2 |       /      2  2              |
                                         ||log\- 2*x*a  + 2*\/  -a  *\/  1 - a *x  /   x*\/  1 - a *x         2     |
                                         ||----------------------------------------- - ----------------  for a  != 0|
                                         ||                      _____                          2                   |
                                         ||                 2   /   2                        2*a                    |
                                       a*|<              2*a *\/  -a                                                |
                                         ||                                                                         |
                                         ||                              3                                          |
                                         ||                             x                                           |
                                         ||                             --                                otherwise |
  /                      2               ||                             3                                           |
 |                      x *asin(a*x)     \\                                                                         /
 | x*asin(a*x) dx = C + ------------ - ------------------------------------------------------------------------------
 |                           2                                               2                                       
/

$${{x^2\,\arcsin \left(a\,x\right)}\over{2}}-{{a\,\left({{\arcsin \left(\left| a\right| \,x\right)}\over{2\,a^2\,\left| a\right| }}-{{ x\,\sqrt{1-a^2\,x^2}}\over{2\,a^2}}\right)}\over{2}}$$

Simplify

The answer [src]

/                       ________                                  
|                      /      2                                   
|asin(a)   asin(a)   \/  1 - a                                    
|------- - ------- + -----------  for And(a > -oo, a < oo, a != 0)
<   2           2        4*a                                      
|            4*a                                                  
|                                                                 
|               0                            otherwise            
\

$${{-\arcsin \left| a\right| +2\,a\,\arcsin a\,\left| a\right| + \sqrt{1-a^2}\,\left| a\right| }\over{4\,a\,\left| a\right| }}$$

/                       ________                                  
|                      /      2                                   
|asin(a)   asin(a)   \/  1 - a                                    
|------- - ------- + -----------  for And(a > -oo, a < oo, a != 0)
<   2           2        4*a                                      
|            4*a                                                  
|                                                                 
|               0                            otherwise            
\

$$\begin{cases} \frac{\operatorname{asin}{\left(a \right)}}{2} + \frac{\sqrt{- a^{2} + 1}}{4 a} - \frac{\operatorname{asin}{\left(a \right)}}{4 a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\0 & \text{otherwise} \end{cases}$$

Simplify

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

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Identical expressions

Integral of xarcsinax dx

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The solution