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Identical expressions
(arccos(3x))^ two /(sqrt(one -9x^ two))
(arc co sinus of e of (3x)) squared divide by ( square root of (1 minus 9x squared ))
(arc co sinus of e of (3x)) to the power of two divide by ( square root of (one minus 9x to the power of two))
(arccos(3x))^2/(√(1-9x^2))
(arccos(3x))2/(sqrt(1-9x2))
arccos3x2/sqrt1-9x2
(arccos(3x))²/(sqrt(1-9x²))
(arccos(3x)) to the power of 2/(sqrt(1-9x to the power of 2))
arccos3x^2/sqrt1-9x^2
(arccos(3x))^2 divide by (sqrt(1-9x^2))
(arccos(3x))^2/(sqrt(1-9x^2))dx
Similar expressions
x+(arccos3x)^2/sqrt(1-9x^2)
(arccos(3x))^2/(sqrt(1+9x^2))

Solving integrals/ (arccos(3x))^2/(sqrt(1-9x^2))

Integral of (arccos(3x))^2/(sqrt(1-9x^2)) dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |        2         
 |    acos (3*x)    
 |  ------------- dx
 |     __________   
 |    /        2    
 |  \/  1 - 9*x     
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{\operatorname{acos}^{2}{\left(3 x \right)}}{\sqrt{1 - 9 x^{2}}}\, dx$$

Integral(acos(3*x)^2/sqrt(1 - 9*x^2), (x, 0, 1))

Detail solution

Let $u = \operatorname{acos}{\left(3 x \right)}$ .

Then let $du = - \frac{3 dx}{\sqrt{1 - 9 x^{2}}}$ and substitute $- \frac{du}{3}$ :

$\int \left(- \frac{u^{2}}{3}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u^{2}\, du = - \frac{\int u^{2}\, du}{3}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{2}\, du = \frac{u^{3}}{3}$
  So, the result is: $- \frac{u^{3}}{9}$
Now substitute $u$ back in:

$- \frac{\operatorname{acos}^{3}{\left(3 x \right)}}{9}$
Add the constant of integration:

$- \frac{\operatorname{acos}^{3}{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer is:

$- \frac{\operatorname{acos}^{3}{\left(3 x \right)}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                                  
 |       2                    3     
 |   acos (3*x)           acos (3*x)
 | ------------- dx = C - ----------
 |    __________              9     
 |   /        2                     
 | \/  1 - 9*x                      
 |                                  
/

$$\int \frac{\operatorname{acos}^{2}{\left(3 x \right)}}{\sqrt{1 - 9 x^{2}}}\, dx = C - \frac{\operatorname{acos}^{3}{\left(3 x \right)}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      3        3
  acos (3)   pi 
- -------- + ---
     9        72

$$\frac{\pi^{3}}{72} - \frac{\operatorname{acos}^{3}{\left(3 \right)}}{9}$$

      3        3
  acos (3)   pi 
- -------- + ---
     9        72

$$\frac{\pi^{3}}{72} - \frac{\operatorname{acos}^{3}{\left(3 \right)}}{9}$$

-acos(3)^3/9 + pi^3/72

Numerical answer [src]

(0.430059689244218 + 0.608241744480679j)

(0.430059689244218 + 0.608241744480679j)

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

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Identical expressions

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Integral of (arccos(3x))^2/(sqrt(1-9x^2)) dx

Limits of integration:

The graph:

Piecewise:

The solution