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tg^3xdx/cos^2x
Solving integrals/ tg^3xdx/cos^2x

Integral of tg^3xdx/cos^2x dx

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∫01tan⁡3(x)1⋅1cos⁡2(x) dx\int\limits_{0}^{1} \tan^{3}{\left(x \right)} 1 \cdot \frac{1}{\cos^{2}{\left(x \right)}}\, dx 
Integral(tan(x)^3*1/cos(x)^2, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    tan⁡3(x)sec⁡2(x)=(sec⁡2(x)−1)tan⁡(x)sec⁡2(x)\tan^{3}{\left(x \right)} \sec^{2}{\left(x \right)} = \left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Let u=sec⁡2(x)u = \sec^{2}{\left(x \right)}.

      Then let du=2tan⁡(x)sec⁡2(x)dxdu = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx and substitute dudu:

      ∫(u2−12) du\int \left(\frac{u}{2} - \frac{1}{2}\right)\, du

      1. Integrate term-by-term:

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

          ∫u2 du=∫u du2\int \frac{u}{2}\, du = \frac{\int u\, du}{2}

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u du=u22\int u\, du = \frac{u^{2}}{2}

          So, the result is: u24\frac{u^{2}}{4}

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

          ∫(−12) du=−u2\int \left(- \frac{1}{2}\right)\, du = - \frac{u}{2}

        The result is: u24−u2\frac{u^{2}}{4} - \frac{u}{2}

      Now substitute uu back in:

      sec⁡4(x)4−sec⁡2(x)2\frac{\sec^{4}{\left(x \right)}}{4} - \frac{\sec^{2}{\left(x \right)}}{2}

    Method #2

    1. Rewrite the integrand:

      (sec⁡2(x)−1)tan⁡(x)sec⁡2(x)=tan⁡(x)sec⁡4(x)−tan⁡(x)sec⁡2(x)\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)} = \tan{\left(x \right)} \sec^{4}{\left(x \right)} - \tan{\left(x \right)} \sec^{2}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=sec⁡4(x)u = \sec^{4}{\left(x \right)}.

        Then let du=4tan⁡(x)sec⁡4(x)dxdu = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx and substitute du4\frac{du}{4}:

        ∫116 du\int \frac{1}{16}\, du

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

          ∫14 du=∫1 du4\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}

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

            ∫1 du=u\int 1\, du = u

          So, the result is: u4\frac{u}{4}

        Now substitute uu back in:

        sec⁡4(x)4\frac{\sec^{4}{\left(x \right)}}{4}

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

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

        1. Let u=sec⁡2(x)u = \sec^{2}{\left(x \right)}.

          Then let du=2tan⁡(x)sec⁡2(x)dxdu = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx and substitute du2\frac{du}{2}:

          ∫14 du\int \frac{1}{4}\, du

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

            ∫12 du=∫1 du2\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u2\frac{u}{2}

          Now substitute uu back in:

          sec⁡2(x)2\frac{\sec^{2}{\left(x \right)}}{2}

        So, the result is: −sec⁡2(x)2- \frac{\sec^{2}{\left(x \right)}}{2}

      The result is: sec⁡4(x)4−sec⁡2(x)2\frac{\sec^{4}{\left(x \right)}}{4} - \frac{\sec^{2}{\left(x \right)}}{2}

    Method #3

    1. Rewrite the integrand:

      (sec⁡2(x)−1)tan⁡(x)sec⁡2(x)=tan⁡(x)sec⁡4(x)−tan⁡(x)sec⁡2(x)\left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)} \sec^{2}{\left(x \right)} = \tan{\left(x \right)} \sec^{4}{\left(x \right)} - \tan{\left(x \right)} \sec^{2}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=sec⁡4(x)u = \sec^{4}{\left(x \right)}.

        Then let du=4tan⁡(x)sec⁡4(x)dxdu = 4 \tan{\left(x \right)} \sec^{4}{\left(x \right)} dx and substitute du4\frac{du}{4}:

        ∫116 du\int \frac{1}{16}\, du

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

          ∫14 du=∫1 du4\int \frac{1}{4}\, du = \frac{\int 1\, du}{4}

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

            ∫1 du=u\int 1\, du = u

          So, the result is: u4\frac{u}{4}

        Now substitute uu back in:

        sec⁡4(x)4\frac{\sec^{4}{\left(x \right)}}{4}

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

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

        1. Let u=sec⁡2(x)u = \sec^{2}{\left(x \right)}.

          Then let du=2tan⁡(x)sec⁡2(x)dxdu = 2 \tan{\left(x \right)} \sec^{2}{\left(x \right)} dx and substitute du2\frac{du}{2}:

          ∫14 du\int \frac{1}{4}\, du

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

            ∫12 du=∫1 du2\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}

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

              ∫1 du=u\int 1\, du = u

            So, the result is: u2\frac{u}{2}

          Now substitute uu back in:

          sec⁡2(x)2\frac{\sec^{2}{\left(x \right)}}{2}

        So, the result is: −sec⁡2(x)2- \frac{\sec^{2}{\left(x \right)}}{2}

      The result is: sec⁡4(x)4−sec⁡2(x)2\frac{\sec^{4}{\left(x \right)}}{4} - \frac{\sec^{2}{\left(x \right)}}{2}

  3. Now simplify:

    (sec⁡2(x)−2)sec⁡2(x)4\frac{\left(\sec^{2}{\left(x \right)} - 2\right) \sec^{2}{\left(x \right)}}{4}

  4. Add the constant of integration:

    (sec⁡2(x)−2)sec⁡2(x)4+constant\frac{\left(\sec^{2}{\left(x \right)} - 2\right) \sec^{2}{\left(x \right)}}{4}+ \mathrm{constant}

The answer is:

(sec⁡2(x)−2)sec⁡2(x)4+constant\frac{\left(\sec^{2}{\left(x \right)} - 2\right) \sec^{2}{\left(x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                            
 |                               2         4   
 |    3         1             sec (x)   sec (x)
 | tan (x)*1*------- dx = C - ------- + -------
 |              2                2         4   
 |           cos (x)                           
 |                                             
/
2 sin⁡2x−14 sin⁡4x−8 sin⁡2x+4{{2\,\sin ^2x-1}\over{4\,\sin ^4x-8\,\sin ^2x+4}}
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0.001.000.100.200.300.400.500.600.700.800.9020-10
Plot the graph f(x)
The answer [src] 
             2   
1   1 - 2*cos (1)
- + -------------
4          4     
      4*cos (1)
−14 sin⁡41−8 sin⁡21+4+sin⁡212 sin⁡41−4 sin⁡21+2+14-{{1}\over{4\,\sin ^41-8\,\sin ^21+4}}+{{\sin ^21}\over{2\,\sin ^41 -4\,\sin ^21+2}}+{{1}\over{4}} 
=
= 
             2   
1   1 - 2*cos (1)
- + -------------
4          4     
      4*cos (1)
14+1−2cos⁡2(1)4cos⁡4(1)\frac{1}{4} + \frac{1 - 2 \cos^{2}{\left(1 \right)}}{4 \cos^{4}{\left(1 \right)}}
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Numerical answer [src] 
1.47078538753166
1.47078538753166
The graph
Integral of tg^3xdx/cos^2x dx

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

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