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Identical expressions
arccos^ three *(5x)/√ one − two 5x^2
arc co sinus of e of cubed multiply by (5x) divide by √1−25x squared
arc co sinus of e of to the power of three multiply by (5x) divide by √ one − two 5x squared
arccos3*(5x)/√1−25x2
arccos3*5x/√1−25x2
arccos³*(5x)/√1−25x²
arccos to the power of 3*(5x)/√1−25x to the power of 2
arccos^3(5x)/√1−25x^2
arccos3(5x)/√1−25x2
arccos35x/√1−25x2
arccos^35x/√1−25x^2
arccos^3*(5x) divide by √1−25x^2
arccos^3*(5x)/√1−25x^2dx

Solving integrals/ arccos^3*(5x)/√1−25x^2

Integral of arccos^3*(5x)/√1−25x^2 dx

The solution

You have entered [src]

  1                        
  /                        
 |                         
 |  /    3             \   
 |  |acos (5*x)       2|   
 |  |---------- - 25*x | dx
 |  |    ___           |   
 |  \  \/ 1            /   
 |                         
/                          
0

$$\int\limits_{0}^{1} \left(- 25 x^{2} + \frac{\operatorname{acos}^{3}{\left(5 x \right)}}{\sqrt{1}}\right)\, dx$$

Integral(acos(5*x)^3/sqrt(1) - 25*x^2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- 25 x^{2}\right)\, dx = - 25 \int x^{2}\, dx$
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  So, the result is: $- \frac{25 x^{3}}{3}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\operatorname{acos}^{3}{\left(5 x \right)}}{\sqrt{1}}\, dx = \int \operatorname{acos}^{3}{\left(5 x \right)}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}$
  So, the result is: $x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}$
The result is: $- \frac{25 x^{3}}{3} + x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}$
Add the constant of integration:

$- \frac{25 x^{3}}{3} + x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}+ \mathrm{constant}$

The answer is:

$- \frac{25 x^{3}}{3} + x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                   
 |                                            ___________                                       ___________           
 | /    3             \              3       /         2                                       /         2      2     
 | |acos (5*x)       2|          25*x    6*\/  1 - 25*x           3                        3*\/  1 - 25*x  *acos (5*x)
 | |---------- - 25*x | dx = C - ----- + ---------------- + x*acos (5*x) - 6*x*acos(5*x) - ---------------------------
 | |    ___           |            3            5                                                       5             
 | \  \/ 1            /                                                                                               
 |                                                                                                                    
/

$$\int \left(- 25 x^{2} + \frac{\operatorname{acos}^{3}{\left(5 x \right)}}{\sqrt{1}}\right)\, dx = C - \frac{25 x^{3}}{3} + x \operatorname{acos}^{3}{\left(5 x \right)} - 6 x \operatorname{acos}{\left(5 x \right)} - \frac{3 \sqrt{1 - 25 x^{2}} \operatorname{acos}^{2}{\left(5 x \right)}}{5} + \frac{6 \sqrt{1 - 25 x^{2}}}{5}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                   2          ___         ___     2   
  143       3                  3*pi    12*I*\/ 6    6*I*\/ 6 *acos (5)
- --- + acos (5) - 6*acos(5) + ----- + ---------- - ------------------
   15                            20        5                5

$$- \frac{143}{15} + \frac{3 \pi^{2}}{20} - 6 \operatorname{acos}{\left(5 \right)} + \operatorname{acos}^{3}{\left(5 \right)} + \frac{12 \sqrt{6} i}{5} - \frac{6 \sqrt{6} i \operatorname{acos}^{2}{\left(5 \right)}}{5}$$

                                   2          ___         ___     2   
  143       3                  3*pi    12*I*\/ 6    6*I*\/ 6 *acos (5)
- --- + acos (5) - 6*acos(5) + ----- + ---------- - ------------------
   15                            20        5                5

-143/15 + acos(5)^3 - 6*acos(5) + 3*pi^2/20 + 12*i*sqrt(6)/5 - 6*i*sqrt(6)*acos(5)^2/5

Numerical answer [src]

(-8.05288765723817 - 4.47590456084446j)

(-8.05288765723817 - 4.47590456084446j)

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

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Identical expressions

Integral of arccos^3*(5x)/√1−25x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution