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Identical expressions
(3tg^ two (x)- fifty)/(2tg(x)+ seven)
(3tg squared (x) minus 50) divide by (2tg(x) plus 7)
(3tg to the power of two (x) minus fifty) divide by (2tg(x) plus seven)
(3tg2(x)-50)/(2tg(x)+7)
3tg2x-50/2tgx+7
(3tg²(x)-50)/(2tg(x)+7)
(3tg to the power of 2(x)-50)/(2tg(x)+7)
3tg^2x-50/2tgx+7
(3tg^2(x)-50) divide by (2tg(x)+7)
(3tg^2(x)-50)/(2tg(x)+7)dx
Similar expressions
(3tg^2(x)-50)/(2tg(x)-7)
(3tg^2(x)+50)/(2tg(x)+7)

Solving integrals/ (3tg^2(x)-50)/(2tg(x)+7)

Integral of (3tg^2(x)-50)/(2tg(x)+7) dx

The solution

You have entered [src]

       0                       
       /                       
      |                        
      |            2           
      |       3*tan (x) - 50   
      |       -------------- dx
      |        2*tan(x) + 7    
      |                        
     /                         
     /  ____\                  
     |\/ 10 |                  
-acos|------|                  
     \  10  /

$$\int\limits_{- \operatorname{acos}{\left(\frac{\sqrt{10}}{10} \right)}}^{0} \frac{3 \tan^{2}{\left(x \right)} - 50}{2 \tan{\left(x \right)} + 7}\, dx$$

Integral((3*tan(x)^2 - 50)/(2*tan(x) + 7), (x, -acos(sqrt(10)/10), 0))

Detail solution

Rewrite the integrand:

$\frac{3 \tan^{2}{\left(x \right)} - 50}{2 \tan{\left(x \right)} + 7} = \frac{3 \tan^{2}{\left(x \right)}}{2 \tan{\left(x \right)} + 7} - \frac{50}{2 \tan{\left(x \right)} + 7}$
Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 \tan^{2}{\left(x \right)}}{2 \tan{\left(x \right)} + 7}\, dx = 3 \int \frac{\tan^{2}{\left(x \right)}}{2 \tan{\left(x \right)} + 7}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{7 x}{53} + \frac{49 \log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{106} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{53}$
  So, the result is: $- \frac{21 x}{53} + \frac{147 \log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{106} + \frac{3 \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{53}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{50}{2 \tan{\left(x \right)} + 7}\right)\, dx = - 50 \int \frac{1}{2 \tan{\left(x \right)} + 7}\, dx$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $\frac{7 x}{53} + \frac{2 \log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{53} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{53}$
  So, the result is: $- \frac{350 x}{53} - \frac{100 \log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{53} + \frac{50 \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{53}$
The result is: $- 7 x - \frac{\log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{2} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}$
Now simplify:

$- 7 x - \frac{\log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{2} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}$
Add the constant of integration:

$- 7 x - \frac{\log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{2} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}+ \mathrm{constant}$

The answer is:

$- 7 x - \frac{\log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{2} + \log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                  
 |                                                                   
 |      2                                                            
 | 3*tan (x) - 50                log(7/2 + tan(x))      /       2   \
 | -------------- dx = C - 7*x - ----------------- + log\1 + tan (x)/
 |  2*tan(x) + 7                         2                           
 |                                                                   
/

$$\int \frac{3 \tan^{2}{\left(x \right)} - 50}{2 \tan{\left(x \right)} + 7}\, dx = C - 7 x - \frac{\log{\left(\tan{\left(x \right)} + \frac{7}{2} \right)}}{2} + \log{\left(\tan^{2}{\left(x \right)} + 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                 /  ____\                    
                 |\/ 10 |   log(2)   log(7/2)
-log(10) - 7*acos|------| - ------ - --------
                 \  10  /     2         2

$$- 7 \operatorname{acos}{\left(\frac{\sqrt{10}}{10} \right)} - \log{\left(10 \right)} - \frac{\log{\left(\frac{7}{2} \right)}}{2} - \frac{\log{\left(2 \right)}}{2}$$

                 /  ____\                    
                 |\/ 10 |   log(2)   log(7/2)
-log(10) - 7*acos|------| - ------ - --------
                 \  10  /     2         2

$$- 7 \operatorname{acos}{\left(\frac{\sqrt{10}}{10} \right)} - \log{\left(10 \right)} - \frac{\log{\left(\frac{7}{2} \right)}}{2} - \frac{\log{\left(2 \right)}}{2}$$

-log(10) - 7*acos(sqrt(10)/10) - log(2)/2 - log(7/2)/2

Numerical answer [src]

-12.0188605743095

-12.0188605743095

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

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Identical expressions

Similar expressions

Integral of (3tg^2(x)-50)/(2tg(x)+7) dx

Limits of integration:

The graph:

Piecewise:

The solution