Mister Exam

Other calculators


sin^3x/cos^4x

Integral of sin^3x/cos^4x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |     3      
 |  sin (x)   
 |  ------- dx
 |     4      
 |  cos (x)   
 |            
/             
0             
$$\int\limits_{0}^{1} \frac{\sin^{3}{\left(x \right)}}{\cos^{4}{\left(x \right)}}\, dx$$
Detail solution
  1. Rewrite the integrand:

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. 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:

      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 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:

      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 - ------ + ---------
 |    4             cos(x)        3   
 | cos (x)                   3*cos (x)
 |                                    
/                                     
$$-{{3\,\cos ^2x-1}\over{3\,\cos ^3x}}$$
The graph
The answer [src]
             2   
2   1 - 3*cos (1)
- + -------------
3          3     
      3*cos (1)  
$$-{{1}\over{\cos 1}}+{{1}\over{3\,\cos ^31}}+{{2}\over{3}}$$
=
=
             2   
2   1 - 3*cos (1)
- + -------------
3          3     
      3*cos (1)  
$$\frac{- 3 \cos^{2}{\left(1 \right)} + 1}{3 \cos^{3}{\left(1 \right)}} + \frac{2}{3}$$
Numerical answer [src]
0.92918564057767
0.92918564057767
The graph
Integral of sin^3x/cos^4x dx

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