Integral of ln(6cosx) dx

The solution

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 |  log(6*cos(x)) dx
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$$\int\limits_{0}^{1} \log{\left(6 \cos{\left(x \right)} \right)}\, dx$$

Integral(log(6*cos(x)), (x, 0, 1))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(6 \cos{\left(x \right)} \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$ .

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:

$\int \left(- \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)\, dx = - \int \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}}\, dx$
So, the result is: $- \int \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}}\, dx$
Now simplify:

$x \log{\left(6 \cos{\left(x \right)} \right)} + \int x \tan{\left(x \right)}\, dx$
Add the constant of integration:

$x \log{\left(6 \cos{\left(x \right)} \right)} + \int x \tan{\left(x \right)}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(6 \cos{\left(x \right)} \right)} + \int x \tan{\left(x \right)}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                                              /           
  /                                          |            
 |                                           | x*sin(x)   
 | log(6*cos(x)) dx = C + x*log(6*cos(x)) +  | -------- dx
 |                                           |  cos(x)    
/                                            |            
                                            /

$$\int \log{\left(6 \cos{\left(x \right)} \right)}\, dx = C + x \log{\left(6 \cos{\left(x \right)} \right)} + \int \frac{x \sin{\left(x \right)}}{\cos{\left(x \right)}}\, dx$$

Simplify

The answer [src]

  1                 
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 |  log(6*cos(x)) dx
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0

$$\int\limits_{0}^{1} \log{\left(6 \cos{\left(x \right)} \right)}\, dx$$

  1                 
  /                 
 |                  
 |  log(6*cos(x)) dx
 |                  
/                   
0

$$\int\limits_{0}^{1} \log{\left(6 \cos{\left(x \right)} \right)}\, dx$$

Integral(log(6*cos(x)), (x, 0, 1))

Numerical answer [src]

1.60422130020722

1.60422130020722

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

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Identical expressions

Integral of ln(6cosx) dx

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Piecewise:

The solution