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tag^5xsec^2xdx
Solving integrals/ tag^5xsec^2xdx

Integral of tag^5xsec^2xdx dx

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The solution

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∫01tan⁡5(x)sec⁡2(x)1 dx\int\limits_{0}^{1} \tan^{5}{\left(x \right)} \sec^{2}{\left(x \right)} 1\, dx 
Integral(tan(x)^5*sec(x)^2*1, (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

      Then let du=(tan⁡2(x)+1)dxdu = \left(\tan^{2}{\left(x \right)} + 1\right) dx and substitute dudu:

      ∫u5 du\int u^{5}\, du

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

        ∫u5 du=u66\int u^{5}\, du = \frac{u^{6}}{6}

      Now substitute uu back in:

      tan⁡6(x)6\frac{\tan^{6}{\left(x \right)}}{6}

    Method #2

    1. Rewrite the integrand:

      tan⁡5(x)sec⁡2(x)1=(sec⁡2(x)−1)2tan⁡(x)sec⁡2(x)\tan^{5}{\left(x \right)} \sec^{2}{\left(x \right)} 1 = \left(\sec^{2}{\left(x \right)} - 1\right)^{2} \tan{\left(x \right)} \sec^{2}{\left(x \right)}

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

      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}:

      ∫u24 du\int \frac{u^{2}}{4}\, du

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

        ∫u22 du=∫u2 du2\int \frac{u^{2}}{2}\, du = \frac{\int u^{2}\, du}{2}

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

          ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

        So, the result is: u36\frac{u^{3}}{6}

      Now substitute uu back in:

      (sec⁡2(x)−1)36\frac{\left(\sec^{2}{\left(x \right)} - 1\right)^{3}}{6}

  2. Add the constant of integration:

    tan⁡6(x)6+constant\frac{\tan^{6}{\left(x \right)}}{6}+ \mathrm{constant}

The answer is:

tan⁡6(x)6+constant\frac{\tan^{6}{\left(x \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                  
 |                               6   
 |    5       2               tan (x)
 | tan (x)*sec (x)*1 dx = C + -------
 |                               6   
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tan⁡6x6{{\tan ^6x}\over{6}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90050
Plot the graph f(x)
The answer [src] 
                4           2   
  1   -1 - 3*cos (1) + 3*cos (1)
- - - --------------------------
  6                6            
              6*cos (1)
−16 sin⁡61−18 sin⁡41+18 sin⁡21−6−sin⁡412 sin⁡61−6 sin⁡41+6 sin⁡21−2+sin⁡212 sin⁡61−6 sin⁡41+6 sin⁡21−2−16-{{1}\over{6\,\sin ^61-18\,\sin ^41+18\,\sin ^21-6}}-{{\sin ^41 }\over{2\,\sin ^61-6\,\sin ^41+6\,\sin ^21-2}}+{{\sin ^21}\over{2\, \sin ^61-6\,\sin ^41+6\,\sin ^21-2}}-{{1}\over{6}} 
=
= 
                4           2   
  1   -1 - 3*cos (1) + 3*cos (1)
- - - --------------------------
  6                6            
              6*cos (1)
−16−−1−3cos⁡4(1)+3cos⁡2(1)6cos⁡6(1)- \frac{1}{6} - \frac{-1 - 3 \cos^{4}{\left(1 \right)} + 3 \cos^{2}{\left(1 \right)}}{6 \cos^{6}{\left(1 \right)}}
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Numerical answer [src] 
2.37827842589157
2.37827842589157
The graph
Integral of tag^5xsec^2xdx dx

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

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