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Solving integrals/ secx

Integral of secx dx

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The solution

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

  1. Rewrite the integrand:

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

  2. Let u=tan⁡(x)+sec⁡(x)u = \tan{\left(x \right)} + \sec{\left(x \right)}.

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

    ∫1u du\int \frac{1}{u}\, du

    1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

    Now substitute uu back in:

    log⁡(tan⁡(x)+sec⁡(x))\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}

  3. Add the constant of integration:

    log⁡(tan⁡(x)+sec⁡(x))+constant\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}+ \mathrm{constant}

The answer is:

log⁡(tan⁡(x)+sec⁡(x))+constant\log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫sec⁡(x) dx=C+log⁡(tan⁡(x)+sec⁡(x))\int \sec{\left(x \right)}\, dx = C + \log{\left(\tan{\left(x \right)} + \sec{\left(x \right)} \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.02.0
Plot the graph f(x)
The answer [src] 
log(1 + sin(1))   log(1 - sin(1))
--------------- - ---------------
       2                 2
log⁡(sin⁡(1)+1)2−log⁡(1−sin⁡(1))2\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} 
=
= 
log(1 + sin(1))   log(1 - sin(1))
--------------- - ---------------
       2                 2
log⁡(sin⁡(1)+1)2−log⁡(1−sin⁡(1))2\frac{\log{\left(\sin{\left(1 \right)} + 1 \right)}}{2} - \frac{\log{\left(1 - \sin{\left(1 \right)} \right)}}{2} 
log(1 + sin(1))/2 - log(1 - sin(1))/2
Numerical answer [src] 
1.22619117088352
1.22619117088352
The graph
Integral of secx dx

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

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