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sec^2xtanx
Solving integrals/ sec^2xtanx

Integral of sec^2xtanx dx

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The solution

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

  1. 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 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}

    Method #2

    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:

      ∫u du\int u\, du

      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}

      Now substitute uu back in:

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

    Method #3

    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:

      ∫u du\int u\, du

      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}

      Now substitute uu back in:

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

  2. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                               
 |                            2   
 |    2                    sec (x)
 | sec (x)*tan(x) dx = C + -------
 |                            2   
/
tan⁡2x2{{\tan ^2x}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
  1       1    
- - + ---------
  2        2   
      2*cos (1)
−12 sin⁡21−2−12-{{1}\over{2\,\sin ^21-2}}-{{1}\over{2}} 
=
= 
  1       1    
- - + ---------
  2        2   
      2*cos (1)
−12+12cos⁡2(1)- \frac{1}{2} + \frac{1}{2 \cos^{2}{\left(1 \right)}}
Simplify
Numerical answer [src] 
1.21275941040738
1.21275941040738
The graph
Integral of sec^2xtanx dx

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

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