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1/(sinx)^4
Solving integrals/ 1/(sinx)^4

Integral of 1/(sinx)^4 dx

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∫011sin⁡4(x) dx\int\limits_{0}^{1} \frac{1}{\sin^{4}{\left(x \right)}}\, dx 
Integral(1/(sin(x)^4), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

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

  2. There are multiple ways to do this integral.

    Method #1

    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 dudu:

      ∫(−u2−1) du\int \left(- u^{2} - 1\right)\, du

      1. Integrate term-by-term:

        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}

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

          ∫(−1) du=−u\int \left(-1\right)\, du = - u

        The result is: −u33−u- \frac{u^{3}}{3} - u

      Now substitute uu back in:

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

    Method #2

    1. Rewrite the integrand:

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

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

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

          ∫u2 du=−∫u2 du\int u^{2}\, 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}

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

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

    Method #3

    1. Rewrite the integrand:

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

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

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

          ∫u2 du=−∫u2 du\int u^{2}\, 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}

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

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                 
 |                              3   
 |    1                      cot (x)
 | ------- dx = C - cot(x) - -------
 |    4                         3   
 | sin (x)                          
 |                                  
/
∫1sin⁡4(x) dx=C−cot⁡3(x)3−cot⁡(x)\int \frac{1}{\sin^{4}{\left(x \right)}}\, dx = C - \frac{\cot^{3}{\left(x \right)}}{3} - \cot{\left(x \right)}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-20000000000000002000000000000000
Plot the graph f(x)
The answer [src] 
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Numerical answer [src] 
7.81431122445857e+56
7.81431122445857e+56
The graph
Integral of 1/(sinx)^4 dx

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

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