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Solving integrals/ tg^2x-ctg^2x

Integral of tg^2x-ctg^2x dx

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The solution

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∫π6π4(tan⁡2(x)−cot⁡2(x)) dx\int\limits_{\frac{\pi}{6}}^{\frac{\pi}{4}} \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)\, dx 
Integral(tan(x)^2 - cot(x)^2, (x, pi/6, pi/4))
Detail solution

  1. Integrate term-by-term:

    1. Rewrite the integrand:

      tan⁡2(x)=sec⁡2(x)−1\tan^{2}{\left(x \right)} = \sec^{2}{\left(x \right)} - 1

    2. Integrate term-by-term:

      1. ∫sec⁡2(x) dx=tan⁡(x)\int \sec^{2}{\left(x \right)}\, dx = \tan{\left(x \right)}

      1. The integral of a constant is the constant times the variable of integration:

        ∫(−1) dx=−x\int \left(-1\right)\, dx = - x

      The result is: −x+tan⁡(x)- x + \tan{\left(x \right)}

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

      ∫(−cot⁡2(x)) dx=−∫cot⁡2(x) dx\int \left(- \cot^{2}{\left(x \right)}\right)\, dx = - \int \cot^{2}{\left(x \right)}\, dx

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

        But the integral is

        −x−cos⁡(x)sin⁡(x)- x - \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

      So, the result is: x+cos⁡(x)sin⁡(x)x + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

    The result is: tan⁡(x)+cos⁡(x)sin⁡(x)\tan{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

  2. Now simplify:

    2sin⁡(2x)\frac{2}{\sin{\left(2 x \right)}}

  3. Add the constant of integration:

    2sin⁡(2x)+constant\frac{2}{\sin{\left(2 x \right)}}+ \mathrm{constant}

The answer is:

2sin⁡(2x)+constant\frac{2}{\sin{\left(2 x \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                            
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 | /   2         2   \          cos(x)         
 | \tan (x) - cot (x)/ dx = C + ------ + tan(x)
 |                              sin(x)         
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∫(tan⁡2(x)−cot⁡2(x)) dx=C+tan⁡(x)+cos⁡(x)sin⁡(x)\int \left(\tan^{2}{\left(x \right)} - \cot^{2}{\left(x \right)}\right)\, dx = C + \tan{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}
Simplify
The graph
0.5250.5500.5750.6000.6250.6500.6750.7000.7250.7500.7755-5
Plot the graph f(x)
The answer [src] 
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    4*\/ 3 
2 - -------
       3
2−4332 - \frac{4 \sqrt{3}}{3} 
=
= 
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    4*\/ 3 
2 - -------
       3
2−4332 - \frac{4 \sqrt{3}}{3} 
2 - 4*sqrt(3)/3
Numerical answer [src] 
-0.309401076758503
-0.309401076758503

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

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