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tg^3x
Solving integrals/ tg^3x

Integral of tg^3x dx

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∫0π4tan⁡3(x) dx\int\limits_{0}^{\frac{\pi}{4}} \tan^{3}{\left(x \right)}\, dx 
Integral(tan(x)^3, (x, 0, pi/4))
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−12u du\int \frac{u - 1}{2 u}\, du

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

        ∫u−1u du=∫u−1u du2\int \frac{u - 1}{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⁡(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}

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

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

            ∫1u du=−∫1u du\int \frac{1}{u}\, 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⁡(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}

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

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

            ∫1u du=−∫1u du\int \frac{1}{u}\, 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      
/
∫tan⁡3(x) dx=C−log⁡(sec⁡2(x))2+sec⁡2(x)2\int \tan^{3}{\left(x \right)}\, dx = C - \frac{\log{\left(\sec^{2}{\left(x \right)} \right)}}{2} + \frac{\sec^{2}{\left(x \right)}}{2}
Simplify
The graph
0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.7502
Plot the graph f(x)
The answer [src] 
       /  ___\
1      |\/ 2 |
- + log|-----|
2      \  2  /
log⁡(22)+12\log{\left(\frac{\sqrt{2}}{2} \right)} + \frac{1}{2} 
=
= 
       /  ___\
1      |\/ 2 |
- + log|-----|
2      \  2  /
log⁡(22)+12\log{\left(\frac{\sqrt{2}}{2} \right)} + \frac{1}{2} 
1/2 + log(sqrt(2)/2)
Numerical answer [src] 
0.153426409720027
0.153426409720027
The graph
Integral of tg^3x dx

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

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