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log(1-acosx)dx

Integral of log(1+acosx)dx dx

The solution

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 POST_GRBEK_SMALL_pi                     
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         |                               
         |          log(1 + acos(x))*1 dx
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        /                                
        0

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

Integral(log(1 + acos(x))*1, (x, 0, POST_GRBEK_SMALL_pi))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(\operatorname{acos}{\left(x \right)} + 1 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

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

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.
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{x}{\sqrt{1 - x^{2}} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\right)\, dx = - \int \frac{x}{\sqrt{1 - x^{2}} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\, dx$
So, the result is: $- \int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\, dx$
Add the constant of integration:

$x \log{\left(\operatorname{acos}{\left(x \right)} + 1 \right)} + \int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(\operatorname{acos}{\left(x \right)} + 1 \right)} + \int \frac{x}{\sqrt{- \left(x - 1\right) \left(x + 1\right)} \left(\operatorname{acos}{\left(x \right)} + 1\right)}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                                                      /                                      
  /                                                  |                                       
 |                                                   |                  x                    
 | log(1 + acos(x))*1 dx = C + x*log(1 + acos(x)) +  | ----------------------------------- dx
 |                                                   |   ___________________                 
/                                                    | \/ -(1 + x)*(-1 + x) *(1 + acos(x))   
                                                     |                                       
                                                    /

$$x\,\log \left| {\rm atan2}\left(\sqrt{1-x}\,\sqrt{x+1} , x\right)+1 \right| -\int {{{\left(2\,x^2\,e^{\log \left(x+1\right)+\log \left(1 -x\right)}+2\,x^4-2\,x^2\right)\,\log \left(e^{\log \left(x+1\right) +\log \left(1-x\right)}+x^2\right)+4\,x\,e^{{{\log \left(x+1\right) }\over{2}}+{{\log \left(1-x\right)}\over{2}}}\,{\rm atan2}\left(e^{ {{\log \left(x+1\right)}\over{2}}+{{\log \left(1-x\right)}\over{2}}} , x\right)+4\,x\,e^{{{\log \left(x+1\right)}\over{2}}+{{\log \left( 1-x\right)}\over{2}}}}\over{\left(\left(x^2-1\right)\,e^{\log \left( x+1\right)+\log \left(1-x\right)}+x^4-x^2\right)\,\left(\log \left(e ^{\log \left(x+1\right)+\log \left(1-x\right)}+x^2\right)\right)^2+ \left(\left(4\,x^2-4\right)\,e^{\log \left(x+1\right)+\log \left(1-x \right)}+4\,x^4-4\,x^2\right)\,{\rm atan2}\left(e^{{{\log \left(x+1 \right)}\over{2}}+{{\log \left(1-x\right)}\over{2}}} , x\right)^2+ \left(\left(8\,x^2-8\right)\,e^{\log \left(x+1\right)+\log \left(1-x \right)}+8\,x^4-8\,x^2\right)\,{\rm atan2}\left(e^{{{\log \left(x+1 \right)}\over{2}}+{{\log \left(1-x\right)}\over{2}}} , x\right)+ \left(4\,x^2-4\right)\,e^{\log \left(x+1\right)+\log \left(1-x \right)}+4\,x^4-4\,x^2}}}{\;dx}$$

The answer [src]

 POST_GRBEK_SMALL_pi                   
          /                            
         |                             
         |          log(1 + acos(x)) dx
         |                             
        /                              
        0

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

 POST_GRBEK_SMALL_pi                   
          /                            
         |                             
         |          log(1 + acos(x)) dx
         |                             
        /                              
        0

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

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

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Integral of log(1+acosx)dx dx

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The solution