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(tgx+ctgx)²
Solving integrals/ (tgx+ctgx)²

Integral of (tgx+ctgx)² dx

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The solution

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∫01(tan⁡(x)+cot⁡(x))2 dx\int\limits_{0}^{1} \left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2}\, dx
Simplify
Detail solution

  1. Rewrite the integrand:

    (tan⁡(x)+cot⁡(x))2=tan⁡2(x)+2tan⁡(x)cot⁡(x)+cot⁡2(x)\left(\tan{\left(x \right)} + \cot{\left(x \right)}\right)^{2} = \tan^{2}{\left(x \right)} + 2 \tan{\left(x \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)}

  2. 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:

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

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

        But the integral is

        xx

      So, the result is: 2x2 x

    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)}}

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

  3. Now simplify:

    tan⁡(x)−1tan⁡(x)\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}

  4. Add the constant of integration:

    tan⁡(x)−1tan⁡(x)+constant\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}+ \mathrm{constant}

The answer is:

tan⁡(x)−1tan⁡(x)+constant\tan{\left(x \right)} - \frac{1}{\tan{\left(x \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                           
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 |                  2          cos(x)         
 | (tan(x) + cot(x))  dx = C - ------ + tan(x)
 |                             sin(x)         
/
tan⁡x−1tan⁡x\tan x-{{1}\over{\tan x}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90200000000-100000000
Plot the graph f(x)
The answer [src] 
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Simplify
Numerical answer [src] 
1.3793236779486e+19
1.3793236779486e+19
The graph
Integral of (tgx+ctgx)² dx

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

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