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dx/(sin^6x)
Solving integrals/ dx/(sin^6x)

Integral of dx/(sin^6x) dx

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∫011⋅1sin⁡6(x) dx\int\limits_{0}^{1} 1 \cdot \frac{1}{\sin^{6}{\left(x \right)}}\, dx 
Integral(1/sin(x)^6, (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    csc⁡6(x)=(cot⁡2(x)+1)2csc⁡2(x)\csc^{6}{\left(x \right)} = \left(\cot^{2}{\left(x \right)} + 1\right)^{2} \csc^{2}{\left(x \right)}

  2. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      (cot⁡2(x)+1)2csc⁡2(x)=cot⁡4(x)csc⁡2(x)+2cot⁡2(x)csc⁡2(x)+csc⁡2(x)\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \csc^{2}{\left(x \right)} = \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)} + 2 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=cot⁡(x)u = \cot{\left(x \right)}.

        Then let du=(−cot⁡2(x)−1)dxdu = \left(- \cot^{2}{\left(x \right)} - 1\right) dx and substitute −du- du:

        ∫u4 du\int u^{4}\, du

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

          ∫(−u4) du=−∫u4 du\int \left(- u^{4}\right)\, du = - \int u^{4}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          So, the result is: −u55- \frac{u^{5}}{5}

        Now substitute uu back in:

        −cot⁡5(x)5- \frac{\cot^{5}{\left(x \right)}}{5}

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

        ∫2cot⁡2(x)csc⁡2(x) dx=2∫cot⁡2(x)csc⁡2(x) dx\int 2 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 2 \int \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx

        1. Let u=cot⁡(x)u = \cot{\left(x \right)}.

          Then let du=(−cot⁡2(x)−1)dxdu = \left(- \cot^{2}{\left(x \right)} - 1\right) dx and substitute −du- du:

          ∫u2 du\int u^{2}\, du

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

            ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            So, the result is: −u33- \frac{u^{3}}{3}

          Now substitute uu back in:

          −cot⁡3(x)3- \frac{\cot^{3}{\left(x \right)}}{3}

        So, the result is: −2cot⁡3(x)3- \frac{2 \cot^{3}{\left(x \right)}}{3}

      1. ∫csc⁡2(x) dx=−cot⁡(x)\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}

      The result is: −cot⁡5(x)5−2cot⁡3(x)3−cot⁡(x)- \frac{\cot^{5}{\left(x \right)}}{5} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}

    Method #2

    1. Rewrite the integrand:

      (cot⁡2(x)+1)2csc⁡2(x)=cot⁡4(x)csc⁡2(x)+2cot⁡2(x)csc⁡2(x)+csc⁡2(x)\left(\cot^{2}{\left(x \right)} + 1\right)^{2} \csc^{2}{\left(x \right)} = \cot^{4}{\left(x \right)} \csc^{2}{\left(x \right)} + 2 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)} + \csc^{2}{\left(x \right)}

    2. Integrate term-by-term:

      1. Let u=cot⁡(x)u = \cot{\left(x \right)}.

        Then let du=(−cot⁡2(x)−1)dxdu = \left(- \cot^{2}{\left(x \right)} - 1\right) dx and substitute −du- du:

        ∫u4 du\int u^{4}\, du

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

          ∫(−u4) du=−∫u4 du\int \left(- u^{4}\right)\, du = - \int u^{4}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

            ∫u4 du=u55\int u^{4}\, du = \frac{u^{5}}{5}

          So, the result is: −u55- \frac{u^{5}}{5}

        Now substitute uu back in:

        −cot⁡5(x)5- \frac{\cot^{5}{\left(x \right)}}{5}

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

        ∫2cot⁡2(x)csc⁡2(x) dx=2∫cot⁡2(x)csc⁡2(x) dx\int 2 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx = 2 \int \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}\, dx

        1. Let u=cot⁡(x)u = \cot{\left(x \right)}.

          Then let du=(−cot⁡2(x)−1)dxdu = \left(- \cot^{2}{\left(x \right)} - 1\right) dx and substitute −du- du:

          ∫u2 du\int u^{2}\, du

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

            ∫(−u2) du=−∫u2 du\int \left(- u^{2}\right)\, du = - \int u^{2}\, du

            1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n≠−1n \neq -1:

              ∫u2 du=u33\int u^{2}\, du = \frac{u^{3}}{3}

            So, the result is: −u33- \frac{u^{3}}{3}

          Now substitute uu back in:

          −cot⁡3(x)3- \frac{\cot^{3}{\left(x \right)}}{3}

        So, the result is: −2cot⁡3(x)3- \frac{2 \cot^{3}{\left(x \right)}}{3}

      1. ∫csc⁡2(x) dx=−cot⁡(x)\int \csc^{2}{\left(x \right)}\, dx = - \cot{\left(x \right)}

      The result is: −cot⁡5(x)5−2cot⁡3(x)3−cot⁡(x)- \frac{\cot^{5}{\left(x \right)}}{5} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}

  3. Add the constant of integration:

    −cot⁡5(x)5−2cot⁡3(x)3−cot⁡(x)+constant- \frac{\cot^{5}{\left(x \right)}}{5} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}

The answer is:

−cot⁡5(x)5−2cot⁡3(x)3−cot⁡(x)+constant- \frac{\cot^{5}{\left(x \right)}}{5} - \frac{2 \cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                               
 |                                  3         5   
 |      1                      2*cot (x)   cot (x)
 | 1*------- dx = C - cot(x) - --------- - -------
 |      6                          3          5   
 |   sin (x)                                      
 |                                                
/
−15 tan⁡4x+10 tan⁡2x+315 tan⁡5x-{{15\,\tan ^4x+10\,\tan ^2x+3}\over{15\,\tan ^5x}}
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The graph
0.001.000.100.200.300.400.500.600.700.800.90-5e235e23
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The answer [src] 
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Numerical answer [src] 
7.0110751903966e+94
7.0110751903966e+94
The graph
Integral of dx/(sin^6x) dx

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

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