Mister Exam

Other calculators


sin^3x/cos^7x

Integral of sin^3x/cos^7x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     3      
 |  sin (x)   
 |  ------- dx
 |     7      
 |  cos (x)   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(x \right)}}{\cos^{7}{\left(x \right)}}\, dx$$
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    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 is when :

            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:

          So, the result is:

        Now substitute back in:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. 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 is when :

            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:

          So, the result is:

        Now substitute back in:

      Now substitute back in:

    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 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. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                      
 |                                       
 |    3                                  
 | sin (x)              1           1    
 | ------- dx = C - --------- + ---------
 |    7                  4           6   
 | cos (x)          4*cos (x)   6*cos (x)
 |                                       
/                                        
$$-{{3\,\sin ^2x-1}\over{12\,\sin ^6x-36\,\sin ^4x+36\,\sin ^2x-12}}$$
The graph
The answer [src]
              2   
1    2 - 3*cos (1)
-- + -------------
12           6    
       12*cos (1) 
$${{1}\over{12\,\sin ^61-36\,\sin ^41+36\,\sin ^21-12}}-{{\sin ^21 }\over{4\,\sin ^61-12\,\sin ^41+12\,\sin ^21-4}}+{{1}\over{12}}$$
=
=
              2   
1    2 - 3*cos (1)
-- + -------------
12           6    
       12*cos (1) 
$$\frac{1}{12} + \frac{- 3 \cos^{2}{\left(1 \right)} + 2}{12 \cos^{6}{\left(1 \right)}}$$
Numerical answer [src]
3.84906381342323
3.84906381342323
The graph
Integral of sin^3x/cos^7x dx

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