Mister Exam

Integral of tan(x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

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 |  tan(x) dx
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01tan(x)dx\int\limits_{0}^{1} \tan{\left(x \right)}\, dx
Integral(tan(x), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

    tan(x)=sin(x)cos(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

  2. Let u=cos(x)u = \cos{\left(x \right)}.

    Then let du=sin(x)dxdu = - \sin{\left(x \right)} dx and substitute du- du:

    1udu\int \frac{1}{u}\, du

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

      (1u)du=1udu\int \left(- \frac{1}{u}\right)\, du = - \int \frac{1}{u}\, du

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

      So, the result is: log(u)- \log{\left(u \right)}

    Now substitute uu back in:

    log(cos(x))- \log{\left(\cos{\left(x \right)} \right)}

  3. Add the constant of integration:

    log(cos(x))+constant- \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}


The answer is:

log(cos(x))+constant- \log{\left(\cos{\left(x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src]
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tan(x)dx=Clog(cos(x))\int \tan{\left(x \right)}\, dx = C - \log{\left(\cos{\left(x \right)} \right)}
The graph
0.001.000.100.200.300.400.500.600.700.800.9002
The answer [src]
-log(cos(1))
log(cos(1))- \log{\left(\cos{\left(1 \right)} \right)}
=
=
-log(cos(1))
log(cos(1))- \log{\left(\cos{\left(1 \right)} \right)}
Numerical answer [src]
0.615626470386014
0.615626470386014
The graph
Integral of tan(x) dx

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