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Solving integrals/ dx/sinxcos^3x

Integral of dx/sinxcos^3x dx

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∫π4π3cos⁡3(x)sin⁡(x) dx\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}\, dx 
Integral(cos(x)^3/sin(x), (x, pi/4, pi/3))
Detail solution

  1. Rewrite the integrand:

    cos⁡3(x)sin⁡(x)=(1−sin⁡2(x))cos⁡(x)sin⁡(x)\frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}} = \frac{\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}

  2. There are multiple ways to do this integral.

    Method #1

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

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

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

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

        ∫u2−1u du=−∫u2−1u du\int \frac{u^{2} - 1}{u}\, du = - \int \frac{u^{2} - 1}{u}\, du

        1. Let u=u2u = u^{2}.

          Then let du=2ududu = 2 u du 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:

          u22−log⁡(u2)2\frac{u^{2}}{2} - \frac{\log{\left(u^{2} \right)}}{2}

        So, the result is: −u22+log⁡(u2)2- \frac{u^{2}}{2} + \frac{\log{\left(u^{2} \right)}}{2}

      Now substitute uu back in:

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

    Method #2

    1. Rewrite the integrand:

      (1−sin⁡2(x))cos⁡(x)sin⁡(x)=−sin⁡2(x)cos⁡(x)−cos⁡(x)sin⁡(x)\frac{\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = - \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)}}{\sin{\left(x \right)}}

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

      ∫(−sin⁡2(x)cos⁡(x)−cos⁡(x)sin⁡(x)) dx=−∫sin⁡2(x)cos⁡(x)−cos⁡(x)sin⁡(x) dx\int \left(- \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)}}{\sin{\left(x \right)}}\right)\, dx = - \int \frac{\sin^{2}{\left(x \right)} \cos{\left(x \right)} - \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

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

        Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

        ∫u2−1u du\int \frac{u^{2} - 1}{u}\, du

        1. Let u=u2u = u^{2}.

          Then let du=2ududu = 2 u du 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:

          u22−log⁡(u2)2\frac{u^{2}}{2} - \frac{\log{\left(u^{2} \right)}}{2}

        Now substitute uu back in:

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

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

    Method #3

    1. Rewrite the integrand:

      (1−sin⁡2(x))cos⁡(x)sin⁡(x)=−sin⁡(x)cos⁡(x)+cos⁡(x)sin⁡(x)\frac{\left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}} = - \sin{\left(x \right)} \cos{\left(x \right)} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}

    2. Integrate term-by-term:

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

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

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

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

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

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

            ∫u du=−∫u du\int 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}

            So, the result is: −u22- \frac{u^{2}}{2}

          Now substitute uu back in:

          −cos⁡2(x)2- \frac{\cos^{2}{\left(x \right)}}{2}

        So, the result is: cos⁡2(x)2\frac{\cos^{2}{\left(x \right)}}{2}

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

        Then let du=cos⁡(x)dxdu = \cos{\left(x \right)} dx and substitute dudu:

        ∫1u du\int \frac{1}{u}\, du

        1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

        Now substitute uu back in:

        log⁡(sin⁡(x))\log{\left(\sin{\left(x \right)} \right)}

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                                       
 |                                        
 |    3                /   2   \      2   
 | cos (x)          log\sin (x)/   sin (x)
 | ------- dx = C + ------------ - -------
 |  sin(x)               2            2   
 |                                        
/
∫cos⁡3(x)sin⁡(x) dx=C+log⁡(sin⁡2(x))2−sin⁡2(x)2\int \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}\, dx = C + \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{2} - \frac{\sin^{2}{\left(x \right)}}{2}
Simplify
The graph
0.8000.8250.8500.8750.9000.9250.9500.9751.0001.0251-1
Plot the graph f(x)
The answer [src] 
         /  ___\      /  ___\
  1      |\/ 2 |      |\/ 3 |
- - - log|-----| + log|-----|
  8      \  2  /      \  2  /
log⁡(32)−18−log⁡(22)\log{\left(\frac{\sqrt{3}}{2} \right)} - \frac{1}{8} - \log{\left(\frac{\sqrt{2}}{2} \right)} 
=
= 
         /  ___\      /  ___\
  1      |\/ 2 |      |\/ 3 |
- - - log|-----| + log|-----|
  8      \  2  /      \  2  /
log⁡(32)−18−log⁡(22)\log{\left(\frac{\sqrt{3}}{2} \right)} - \frac{1}{8} - \log{\left(\frac{\sqrt{2}}{2} \right)} 
-1/8 - log(sqrt(2)/2) + log(sqrt(3)/2)
Numerical answer [src] 
0.0777325540540822
0.0777325540540822

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

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