Question

What is the integral of ${\sec ^4}\left( x \right){\tan ^4}\left( x \right)dx$ ?

Answer 1

Here we are going to find the integral value of a given function by using substitution method. In the substitution method we substitute a term in form of a variable and simplify the expression so that we can integrate the function by using standard formulae of integration.

Complete step-by-step solution:
Given $I = \int {{{\sec }^4}\left( x \right){{\tan }^4}\left( x \right)dx}$------------------(1)
When integrating a function that is a product of tangents and secants, it is easy to simply integrate by substitution. This may be confusing.
In equation(1) secant function has power four so we can split the secant function like following,
$I = \int {{{\sec }^2}\left( x \right){{\sec }^2}\left( x \right){{\tan }^4}\left( x \right)dx}$----------------(2)
Now we will make one of the secant function in terms of tangent function so we have only possibility is We know the Pythagorean identity that is ${\sec ^2}x = {\tan ^2}x + 1$ we can substituting this in equation (2) we get,
$I = \int {{{\sec }^2}\left( x \right)\left( {{{\tan }^2}\left( x \right) + 1} \right){{\tan }^4}\left( x \right)dx}$
Now before going to do the substitution method we make sure if the function was suitable to do the method. If not we would make for that.
Here is not so we are going to multiply the terms inside bracket by the term which is in outside bracket, we need to multiply the tangent function inside the bracket, so we get,
$I = \int {{{\sec }^2}\left( x \right)\left( {{{\tan }^6}\left( x \right) + {{\tan }^4}\left( x \right)} \right)dx}$---------------(3)
Now using the substitution method that is let take $u = \tan x$ , $du = {\sec ^2}x\,dx$ , we get,
$I = \int {\left( {{u^6} + {u^4}} \right)\,du}$
Integrating by using the formula $\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C}$ we get,
$I = \left[ {\dfrac{{{u^7}}}{7} + \dfrac{{{u^5}}}{5}} \right] + C$
Now undoing the substitution we get,
$I = \left[ {\dfrac{{{{\tan }^7}x}}{7} + \dfrac{{{{\tan }^5}x}}{5}} \right] + C$
Where $C$ is the arbitrary constant of indefinite integration.
This is the final answer.

Note: The substitution method is used when an integral contains some function and its derivative. In this case, we can set $u$ equal to the function and rewrite the integral in terms of the new variable $u$. This makes the integral easier to solve. Do not forget to express the final answer in terms of the original variable $x$.