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tan^3x
Solving integrals/ tan^3x

Integral of tan^3x dx

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The solution

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∫01tan⁡3(x) dx\int\limits_{0}^{1} \tan^{3}{\left(x \right)}\, dx
Simplify
Detail solution

  1. Rewrite the integrand:

    tan⁡3(x)=(sec⁡2(x)−1)tan⁡(x)\tan^{3}{\left(x \right)} = \left(\sec^{2}{\left(x \right)} - 1\right) \tan{\left(x \right)}

  2. 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}:

      ∫u−14u du\int \frac{u - 1}{4 u}\, du

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

        ∫u−12u du=∫u−1u du2\int \frac{u - 1}{2 u}\, du = \frac{\int \frac{u - 1}{u}\, du}{2}

        1. Rewrite the integrand:

          u−1u=1−1u\frac{u - 1}{u} = 1 - \frac{1}{u}

        2. Integrate term-by-term:

          1. The integral of a constant is the constant times the variable of integration:

            ∫1 du=u\int 1\, du = u

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

            ∫(−1u) du=−∫1u du\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)}

          The result is: u−log⁡(u)u - \log{\left(u \right)}

        So, the result is: u2−log⁡(u)2\frac{u}{2} - \frac{\log{\left(u \right)}}{2}

      Now substitute uu back in:

      −log⁡(sec⁡2(x))2+sec⁡2(x)2- \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{\sec^{2}{\left(x \right)}}{2}

    Method #2

    1. Rewrite the integrand:

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

    2. Integrate term-by-term:

      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}

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

        ∫(−tan⁡(x)) dx=−∫tan⁡(x) dx\int \left(- \tan{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)}\, dx

        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:

          ∫1u du\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=−∫1u du\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)}

        So, the result is: log⁡(cos⁡(x))\log{\left(\cos{\left(x \right)} \right)}

      The result is: log⁡(cos⁡(x))+sec⁡2(x)2\log{\left(\cos{\left(x \right)} \right)} + \frac{\sec^{2}{\left(x \right)}}{2}

    Method #3

    1. Rewrite the integrand:

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

    2. Integrate term-by-term:

      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}

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

        ∫(−tan⁡(x)) dx=−∫tan⁡(x) dx\int \left(- \tan{\left(x \right)}\right)\, dx = - \int \tan{\left(x \right)}\, dx

        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:

          ∫1u du\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=−∫1u du\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)}

        So, the result is: log⁡(cos⁡(x))\log{\left(\cos{\left(x \right)} \right)}

      The result is: log⁡(cos⁡(x))+sec⁡2(x)2\log{\left(\cos{\left(x \right)} \right)} + \frac{\sec^{2}{\left(x \right)}}{2}

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                       
 |                     2         /   2   \
 |    3             sec (x)   log\sec (x)/
 | tan (x) dx = C + ------- - ------------
 |                     2           2      
/
log⁡(sin⁡2x−1)2−12 sin⁡2x−2{{\log \left(\sin ^2x-1\right)}\over{2}}-{{1}\over{2\,\sin ^2x-2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.9005
Plot the graph f(x)
The answer [src] 
  1       1                  
- - + --------- + log(cos(1))
  2        2                 
      2*cos (1)
log⁡(1−sin⁡21)2−12 sin⁡21−2−12{{\log \left(1-\sin ^21\right)}\over{2}}-{{1}\over{2\,\sin ^21-2}}- {{1}\over{2}} 
=
= 
  1       1                  
- - + --------- + log(cos(1))
  2        2                 
      2*cos (1)
log⁡(cos⁡(1))−12+12cos⁡2(1)\log{\left(\cos{\left(1 \right)} \right)} - \frac{1}{2} + \frac{1}{2 \cos^{2}{\left(1 \right)}}
Simplify
Numerical answer [src] 
0.597132940021366
0.597132940021366
The graph
Integral of tan^3x dx

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

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