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Integral of 1/1+acos(x) dx

The solution

You have entered [src]

 2*pi                
   /                 
  |                  
  |  (1 + acos(x)) dx
  |                  
 /                   
 0

$$\int\limits_{0}^{2 \pi} \left(\operatorname{acos}{\left(x \right)} + 1\right)\, dx$$

Integral(1 + acos(x), (x, 0, 2*pi))

Detail solution

Integrate term-by-term:
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \operatorname{acos}{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = - \frac{1}{\sqrt{1 - x^{2}}}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{x}{\sqrt{1 - x^{2}}}\right)\, dx = - \int \frac{x}{\sqrt{1 - x^{2}}}\, dx$
  1. Let $u = 1 - x^{2}$ .
    
    Then let $du = - 2 x dx$ and substitute $- \frac{du}{2}$ :
    
    $\int \left(- \frac{1}{2 \sqrt{u}}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{\sqrt{u}}\, du = - \frac{\int \frac{1}{\sqrt{u}}\, du}{2}$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int \frac{1}{\sqrt{u}}\, du = 2 \sqrt{u}$
      So, the result is: $- \sqrt{u}$
    Now substitute $u$ back in:
    
    $- \sqrt{1 - x^{2}}$
  So, the result is: $\sqrt{1 - x^{2}}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x \operatorname{acos}{\left(x \right)} + x - \sqrt{1 - x^{2}}$
Add the constant of integration:

$x \operatorname{acos}{\left(x \right)} + x - \sqrt{1 - x^{2}}+ \mathrm{constant}$

The answer is:

$x \operatorname{acos}{\left(x \right)} + x - \sqrt{1 - x^{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                              ________            
 |                              /      2             
 | (1 + acos(x)) dx = C + x - \/  1 - x   + x*acos(x)
 |                                                   
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

       ___________                         
      /         2                          
1 - \/  1 - 4*pi   + 2*pi + 2*pi*acos(2*pi)

$$1 + 2 \pi - \sqrt{1 - 4 \pi^{2}} + 2 \pi \operatorname{acos}{\left(2 \pi \right)}$$

       ___________                         
      /         2                          
1 - \/  1 - 4*pi   + 2*pi + 2*pi*acos(2*pi)

$$1 + 2 \pi - \sqrt{1 - 4 \pi^{2}} + 2 \pi \operatorname{acos}{\left(2 \pi \right)}$$

1 - sqrt(1 - 4*pi^2) + 2*pi + 2*pi*acos(2*pi)

Numerical answer [src]

(7.28402542760913 + 9.66052250292926j)

(7.28402542760913 + 9.66052250292926j)

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

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Integral of 1/1+acos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution