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Identical expressions
x*arcsin(x/ three)
x multiply by arc sinus of (x divide by 3)
x multiply by arc sinus of (x divide by three)
xarcsin(x/3)
xarcsinx/3
x*arcsin(x divide by 3)
x*arcsin(x/3)dx

Solving integrals/ x*arcsin(x/3)

Integral of x*arcsin(x/3) dx

The solution

You have entered [src]

  3             
  /             
 |              
 |        /x\   
 |  x*asin|-| dx
 |        \3/   
 |              
/               
0

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

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

Detail solution

Use integration by parts:

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

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

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

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:

$\int \frac{x^{2}}{6 \sqrt{1 - \frac{x^{2}}{9}}}\, dx = \frac{\int \frac{x^{2}}{\sqrt{1 - \frac{x^{2}}{9}}}\, dx}{6}$
So, the result is: $- \frac{3 x \sqrt{1 - \frac{x^{2}}{9}}}{4} + \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{4}$
Now simplify:

$\frac{x^{2} \operatorname{asin}{\left(\frac{x}{3} \right)}}{2} + \frac{x \sqrt{9 - x^{2}}}{4} - \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{4}$
Add the constant of integration:

$\frac{x^{2} \operatorname{asin}{\left(\frac{x}{3} \right)}}{2} + \frac{x \sqrt{9 - x^{2}}}{4} - \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{4}+ \mathrm{constant}$

The answer is:

$\frac{x^{2} \operatorname{asin}{\left(\frac{x}{3} \right)}}{2} + \frac{x \sqrt{9 - x^{2}}}{4} - \frac{9 \operatorname{asin}{\left(\frac{x}{3} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                        ________
                                                       /      2 
  /                         /x\    2     /x\          /      x  
 |                    9*asin|-|   x *asin|-|   3*x*  /   1 - -- 
 |       /x\                \3/          \3/       \/        9  
 | x*asin|-| dx = C - --------- + ---------- + -----------------
 |       \3/              4           2                4        
 |                                                              
/

$${{\arcsin \left({{x}\over{3}}\right)\,x^2}\over{2}}-{{{{27\, \arcsin \left({{x}\over{3}}\right)}\over{2}}-{{9\,x\,\sqrt{1-{{x^2 }\over{9}}}}\over{2}}}\over{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

9*pi
----
 8

$${{9\,\pi}\over{8}}$$

9*pi
----
 8

$$\frac{9 \pi}{8}$$

Simplify

Numerical answer [src]

3.53429173528852

3.53429173528852

The graph

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

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Identical expressions

Integral of x*arcsin(x/3) dx

Limits of integration:

The graph:

Piecewise:

The solution