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Solving integrals/ tan(x)(sec(x))^3

Integral of tan(x)(sec(x))^3 dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                  
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 |  tan(x)*sec (x) dx
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∫01tan⁡(x)sec⁡3(x) dx\int\limits_{0}^{1} \tan{\left(x \right)} \sec^{3}{\left(x \right)}\, dx 
Integral(tan(x)*sec(x)^3, (x, 0, 1))
Detail solution

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

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

    ∫u2 du\int u^{2}\, du

    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}

    Now substitute uu back in:

    sec⁡3(x)3\frac{\sec^{3}{\left(x \right)}}{3}

  2. Add the constant of integration:

    sec⁡3(x)3+constant\frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}

The answer is:

sec⁡3(x)3+constant\frac{\sec^{3}{\left(x \right)}}{3}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                               
 |                            3   
 |           3             sec (x)
 | tan(x)*sec (x) dx = C + -------
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∫tan⁡(x)sec⁡3(x) dx=C+sec⁡3(x)3\int \tan{\left(x \right)} \sec^{3}{\left(x \right)}\, dx = C + \frac{\sec^{3}{\left(x \right)}}{3}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90020
Plot the graph f(x)
The answer [src] 
  1       1    
- - + ---------
  3        3   
      3*cos (1)
−13+13cos⁡3(1)- \frac{1}{3} + \frac{1}{3 \cos^{3}{\left(1 \right)}} 
=
= 
  1       1    
- - + ---------
  3        3   
      3*cos (1)
−13+13cos⁡3(1)- \frac{1}{3} + \frac{1}{3 \cos^{3}{\left(1 \right)}} 
-1/3 + 1/(3*cos(1)^3)
Numerical answer [src] 
1.7800013582586
1.7800013582586

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

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