Mister Exam

Other calculators


tan^3x×sec^3x

Integral of tan^3x×sec^3x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

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

  2. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. 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. Integrate term-by-term:

      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 a constant is the constant times the variable of integration:

          So, the result is:

        Now substitute back in:

      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 a constant is the constant times the variable of integration:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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 a constant is the constant times the variable of integration:

          So, the result is:

        Now substitute back in:

      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 a constant is the constant times the variable of integration:

            So, the result is:

          Now substitute back in:

        So, the result is:

      The result is:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                          
 |                             3         5   
 |    3       3             sec (x)   sec (x)
 | tan (x)*sec (x) dx = C - ------- + -------
 |                             3         5   
/                                            
$$-{{5\,\cos ^2x-3}\over{15\,\cos ^5x}}$$
The graph
The answer [src]
              2   
2    3 - 5*cos (1)
-- + -------------
15           5    
       15*cos (1) 
$$-{{1}\over{3\,\cos ^31}}+{{1}\over{5\,\cos ^51}}+{{2}\over{15}}$$
=
=
              2   
2    3 - 5*cos (1)
-- + -------------
15           5    
       15*cos (1) 
$$\frac{2}{15} + \frac{- 5 \cos^{2}{\left(1 \right)} + 3}{15 \cos^{5}{\left(1 \right)}}$$
Numerical answer [src]
2.36355929817875
2.36355929817875
The graph
Integral of tan^3x×sec^3x dx

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