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x*arcsin(3x)
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2x*arcsin3x

Integral of x*arcsin(3x) dx

The solution

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0

$$\int\limits_{0}^{\frac{1}{3}} x \operatorname{asin}{\left(3 x \right)}\, dx$$

Integral(x*asin(3*x), (x, 0, 1/3))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \operatorname{asin}{\left(3 x \right)}$ and let $\operatorname{dv}{\left(x \right)} = x$ .

Then $\operatorname{du}{\left(x \right)} = \frac{3}{\sqrt{1 - 9 x^{2}}}$ .

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:

$\int \frac{3 x^{2}}{2 \sqrt{1 - 9 x^{2}}}\, dx = \frac{3 \int \frac{x^{2}}{\sqrt{1 - 9 x^{2}}}\, dx}{2}$
So, the result is: $\frac{3 \left(\begin{cases} - \frac{x \sqrt{1 - 9 x^{2}}}{18} + \frac{\operatorname{asin}{\left(3 x \right)}}{54} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}\right)}{2}$
Now simplify:

$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(3 x \right)}}{2} + \frac{x \sqrt{1 - 9 x^{2}}}{12} - \frac{\operatorname{asin}{\left(3 x \right)}}{36} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}$
Add the constant of integration:

$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(3 x \right)}}{2} + \frac{x \sqrt{1 - 9 x^{2}}}{12} - \frac{\operatorname{asin}{\left(3 x \right)}}{36} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}$

The answer is:

$\begin{cases} \frac{x^{2} \operatorname{asin}{\left(3 x \right)}}{2} + \frac{x \sqrt{1 - 9 x^{2}}}{12} - \frac{\operatorname{asin}{\left(3 x \right)}}{36} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}$

The answer (Indefinite) [src]

                          //                 __________                            \               
                          ||                /        2                             |               
                        3*| -1/3, x < 1/3)|    2          
 |                        \\    54             18                                  /   x *asin(3*x)
 | x*asin(3*x) dx = C - ------------------------------------------------------------ + ------------
 |                                                   2                                      2      
/

$$\int x \operatorname{asin}{\left(3 x \right)}\, dx = C + \frac{x^{2} \operatorname{asin}{\left(3 x \right)}}{2} - \frac{3 \left(\begin{cases} - \frac{x \sqrt{1 - 9 x^{2}}}{18} + \frac{\operatorname{asin}{\left(3 x \right)}}{54} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}\right)}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

pi
--
72

$$\frac{\pi}{72}$$

pi
--
72

$$\frac{\pi}{72}$$

pi/72

Numerical answer [src]

0.0436332312998582

0.0436332312998582

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

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Integral of x*arcsin(3x) dx

Limits of integration:

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Piecewise:

The solution