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Integral of tg(4x-2) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  3                
  /                
 |                 
 |  tan(4*x - 2) dx
 |                 
/                  
1                  
$$\int\limits_{1}^{3} \tan{\left(4 x - 2 \right)}\, dx$$
Integral(tan(4*x - 2), (x, 1, 3))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

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

        1. The integral of is .

        So, the result is:

      Now substitute back in:

    Method #2

    1. Let .

      Then let and substitute :

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

        1. Let .

          Then let and substitute :

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

            1. The integral of is .

            So, the result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       
 |                       log(cos(4*x - 2))
 | tan(4*x - 2) dx = C - -----------------
 |                               4        
/                                         
$$\int \tan{\left(4 x - 2 \right)}\, dx = C - \frac{\log{\left(\cos{\left(4 x - 2 \right)} \right)}}{4}$$
The graph
The answer [src]
     /       2   \      /       2    \
  log\1 + tan (2)/   log\1 + tan (10)/
- ---------------- + -----------------
         8                   8        
$$- \frac{\log{\left(1 + \tan^{2}{\left(2 \right)} \right)}}{8} + \frac{\log{\left(\tan^{2}{\left(10 \right)} + 1 \right)}}{8}$$
=
=
     /       2   \      /       2    \
  log\1 + tan (2)/   log\1 + tan (10)/
- ---------------- + -----------------
         8                   8        
$$- \frac{\log{\left(1 + \tan^{2}{\left(2 \right)} \right)}}{8} + \frac{\log{\left(\tan^{2}{\left(10 \right)} + 1 \right)}}{8}$$
-log(1 + tan(2)^2)/8 + log(1 + tan(10)^2)/8
Numerical answer [src]
1.98789651502355
1.98789651502355

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