Question

How do you integrate ${\tan ^4}(x).{\sec ^4}(x).dx$ ?

Answer 1

We need to evaluate $\int {{{\tan }^4}(x).{{\sec }^4}(x).dx}$. Before applying the integration we simplify the given terms inside the integration, that is we simplify ${\tan ^4}(x).{\sec ^4}(x).dx$ using the Pythagorean identity relation between secant and tangent. After simplify we use the substitution method to solve the given problem.

Complete step by step solution:
Given,
$\int {{{\tan }^4}(x).{{\sec }^4}(x).dx}$.
We can write ${\sec ^4}(x)$ as ${\sec ^2}(x).{\sec ^2}(x)$. Then above becomes,
$= \int {{{\tan }^4}(x).{{\sec }^2}(x).{{\sec }^2}(x)dx}$
We have a Pythagorean identity relation between secant and tangent, that is ${\sec ^2}(x) = {\tan ^2}(x) + 1$. Substituting we have,
$= \int {{{\tan }^4}(x).\left( {{{\tan }^2}(x) + 1} \right).{{\sec }^2}(x)dx}$
Multiply tangent inside the parentheses we have,
$= \int {\left( {{{\tan }^6}(x) + {{\tan }^4}(x)} \right).{{\sec }^2}(x)dx}$.
To simplify this easily we put,
$u = \tan (x)$
Differentiating with respect to ‘x’ we have,
$du = {\sec ^2}(x)dx$. Now substituting these value in the given integral we have,
$= \int {\left( {{u^6} + {u^4}} \right)du}$
Now applying the integration with respect to ‘u’ we have,
$= \dfrac{{{u^{6 + 1}}}}{{6 + 1}} + \dfrac{{{u^{4 + 1}}}}{{4 + 1}} + C$.
Where ‘C’ is the integration constant.
$= \dfrac{{{u^7}}}{7} + \dfrac{{{u^5}}}{5} + C$.
But we have to substitute $u = \tan (x)$. We need the answer in terms of ‘x’ only, hence we substitute the value of u in the above equation.
$= \dfrac{{{{\tan }^7}(x)}}{7} + \dfrac{{{{\tan }^5}(x)}}{5} + C$.
Thus we have,
$\Rightarrow \int {{{\tan }^4}(x).{{\sec }^4}(x).dx} = \dfrac{{{{\tan }^7}(x)}}{7} + \dfrac{{{{\tan }^5}(x)}}{5} + C$, where ‘C’ is the integration constant.

Note: In the given problem we have indefinite integral. In an indefinite integral we don’t have lower limit and upper limit. Hence, we have integration constant. But in the definite integral we have lower limit and upper limit. Hence, in the definite integral we don’t have integral constant. We know the integration formula $\int {{x^n}.dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}} + c$. As above we can see that by substituting method we can solve the integration easily. Also don’t get confused with${\sec ^4}(x)$. That power ‘4’ is not multiplied to the angel. That is ${\sec ^4}(x) = {(\sec x)^4}$.