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

Limits of integration:

from to
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The graph:

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Piecewise:

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
  1. Rewrite the integrand:

  2. Integrate term-by-term:

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

      1. Don't know the steps in finding this integral.

        But the integral is

      So, the result is:

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

      1. Don't know the steps in finding this integral.

        But the integral is

      So, the result is:

    The result is:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

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