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Solving integrals/ 1/sin^4x

Integral of 1/sin^4x dx

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∫1221sin⁡4(x) dx\int\limits_{1}^{\frac{\sqrt{2}}{2}} \frac{1}{\sin^{4}{\left(x \right)}}\, dx 
Integral(1/(sin(x)^4), (x, 1, sqrt(2)/2))
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
1.0000.7250.7500.7750.8000.8250.8500.8750.9000.9250.9500.975-1010
Plot the graph f(x)
The answer [src] 
       /  ___\        /  ___\                        
       |\/ 2 |        |\/ 2 |                        
  2*cos|-----|     cos|-----|                        
       \  2  /        \  2  /      cos(1)    2*cos(1)
- ------------ - ------------- + --------- + --------
       /  ___\         /  ___\        3      3*sin(1)
       |\/ 2 |        3|\/ 2 |   3*sin (1)           
  3*sin|-----|   3*sin |-----|                       
       \  2  /         \  2  /
−cos⁡(22)3sin⁡3(22)−2cos⁡(22)3sin⁡(22)+cos⁡(1)3sin⁡3(1)+2cos⁡(1)3sin⁡(1)- \frac{\cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin^{3}{\left(\frac{\sqrt{2}}{2} \right)}} - \frac{2 \cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin{\left(\frac{\sqrt{2}}{2} \right)}} + \frac{\cos{\left(1 \right)}}{3 \sin^{3}{\left(1 \right)}} + \frac{2 \cos{\left(1 \right)}}{3 \sin{\left(1 \right)}} 
=
= 
       /  ___\        /  ___\                        
       |\/ 2 |        |\/ 2 |                        
  2*cos|-----|     cos|-----|                        
       \  2  /        \  2  /      cos(1)    2*cos(1)
- ------------ - ------------- + --------- + --------
       /  ___\         /  ___\        3      3*sin(1)
       |\/ 2 |        3|\/ 2 |   3*sin (1)           
  3*sin|-----|   3*sin |-----|                       
       \  2  /         \  2  /
−cos⁡(22)3sin⁡3(22)−2cos⁡(22)3sin⁡(22)+cos⁡(1)3sin⁡3(1)+2cos⁡(1)3sin⁡(1)- \frac{\cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin^{3}{\left(\frac{\sqrt{2}}{2} \right)}} - \frac{2 \cos{\left(\frac{\sqrt{2}}{2} \right)}}{3 \sin{\left(\frac{\sqrt{2}}{2} \right)}} + \frac{\cos{\left(1 \right)}}{3 \sin^{3}{\left(1 \right)}} + \frac{2 \cos{\left(1 \right)}}{3 \sin{\left(1 \right)}} 
-2*cos(sqrt(2)/2)/(3*sin(sqrt(2)/2)) - cos(sqrt(2)/2)/(3*sin(sqrt(2)/2)^3) + cos(1)/(3*sin(1)^3) + 2*cos(1)/(3*sin(1))
Numerical answer [src] 
-0.974154827019863
-0.974154827019863

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

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