Mister Exam

Other calculators


1/cosx*sin^3x

Integral of 1/cosx*sin^3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                    
  /                    
 |                     
 |      1       3      
 |  1*------*sin (x) dx
 |    cos(x)           
 |                     
/                      
0                      
$$\int\limits_{0}^{1} 1 \cdot \frac{1}{\cos{\left(x \right)}} \sin^{3}{\left(x \right)}\, dx$$
Integral(1*sin(x)^3/cos(x), (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. 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:

    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 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:

    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 .

          So, the result is:

        Now substitute back in:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                
 |                              2         /   2   \
 |     1       3             cos (x)   log\cos (x)/
 | 1*------*sin (x) dx = C + ------- - ------------
 |   cos(x)                     2           2      
 |                                                 
/                                                  
$$-{{\log \left(\sin ^2x-1\right)}\over{2}}-{{\sin ^2x}\over{2}}$$
The graph
The answer [src]
         2                 
  1   cos (1)              
- - + ------- - log(cos(1))
  2      2                 
$$-{{\log \left(1-\sin ^21\right)}\over{2}}-{{\sin ^21}\over{2}}$$
=
=
         2                 
  1   cos (1)              
- - + ------- - log(cos(1))
  2      2                 
$$- \frac{1}{2} + \frac{\cos^{2}{\left(1 \right)}}{2} - \log{\left(\cos{\left(1 \right)} \right)}$$
Numerical answer [src]
0.261589761249229
0.261589761249229
The graph
Integral of 1/cosx*sin^3x dx

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