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Integral of dx/sinxcos^3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
 pi           
 --           
 3            
  /           
 |            
 |     3      
 |  cos (x)   
 |  ------- dx
 |   sin(x)   
 |            
/             
pi            
--            
4             
$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{3}} \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}\, dx$$
Integral(cos(x)^3/sin(x), (x, pi/4, pi/3))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

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

        1. Let .

          Then let and substitute :

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

            1. Rewrite the integrand:

            2. Integrate term-by-term:

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

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

                1. The integral of is .

                So, the result is:

              The result is:

            So, the result is:

          Now substitute back in:

        So, the result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

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

      1. Let .

        Then let and substitute :

        1. Let .

          Then let and substitute :

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

            1. Rewrite the integrand:

            2. Integrate term-by-term:

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

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

                1. The integral of is .

                So, the result is:

              The result is:

            So, the result is:

          Now substitute back in:

        Now substitute back in:

      So, the result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

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

        1. Let .

          Then let and substitute :

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

            1. The integral of is when :

            So, the result is:

          Now substitute back in:

        So, the result is:

      1. Let .

        Then let and substitute :

        1. The integral of is .

        Now substitute back in:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                       
 |                                        
 |    3                /   2   \      2   
 | cos (x)          log\sin (x)/   sin (x)
 | ------- dx = C + ------------ - -------
 |  sin(x)               2            2   
 |                                        
/                                         
$$\int \frac{\cos^{3}{\left(x \right)}}{\sin{\left(x \right)}}\, dx = C + \frac{\log{\left(\sin^{2}{\left(x \right)} \right)}}{2} - \frac{\sin^{2}{\left(x \right)}}{2}$$
The graph
The answer [src]
         /  ___\      /  ___\
  1      |\/ 2 |      |\/ 3 |
- - - log|-----| + log|-----|
  8      \  2  /      \  2  /
$$\log{\left(\frac{\sqrt{3}}{2} \right)} - \frac{1}{8} - \log{\left(\frac{\sqrt{2}}{2} \right)}$$
=
=
         /  ___\      /  ___\
  1      |\/ 2 |      |\/ 3 |
- - - log|-----| + log|-----|
  8      \  2  /      \  2  /
$$\log{\left(\frac{\sqrt{3}}{2} \right)} - \frac{1}{8} - \log{\left(\frac{\sqrt{2}}{2} \right)}$$
-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.