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sin^5x/cos^4x

Integral of sin^5x/cos^4x dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     5      
 |  sin (x)   
 |  ------- dx
 |     4      
 |  cos (x)   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{\sin^{5}{\left(x \right)}}{\cos^{4}{\left(x \right)}}\, dx$$
Integral(sin(x)^5/(cos(x)^4), (x, 0, 1))
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      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 when :

          So, the result is:

        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:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Let .

      Then let and substitute :

      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 when :

          So, the result is:

        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:

        The result is:

      Now substitute back in:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. The integral of sine is negative cosine:

      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 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:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                            
 |                                             
 |    5                                        
 | sin (x)                     2          1    
 | ------- dx = C - cos(x) - ------ + ---------
 |    4                      cos(x)        3   
 | cos (x)                            3*cos (x)
 |                                             
/                                              
$$-{{6\,\cos ^2x-1}\over{3\,\cos ^3x}}-\cos x$$
The graph
The answer [src]
                       2   
8            -1 + 6*cos (1)
- - cos(1) - --------------
3                   3      
               3*cos (1)   
$$-\cos 1-{{2}\over{\cos 1}}+{{1}\over{3\,\cos ^31}}+{{8}\over{3}}$$
=
=
                       2   
8            -1 + 6*cos (1)
- - cos(1) - --------------
3                   3      
               3*cos (1)   
$$- \frac{-1 + 6 \cos^{2}{\left(1 \right)}}{3 \cos^{3}{\left(1 \right)}} - \cos{\left(1 \right)} + \frac{8}{3}$$
Numerical answer [src]
0.538067617028605
0.538067617028605
The graph
Integral of sin^5x/cos^4x dx

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