Find the anti-derivative and simplify.
[tex]\begin{array}{l}
\int \frac{\sec ^2 x}{\tan x \sqrt{\tan ^2 x-1}} d x \\
=[?]^{-1}|[\quad] x|+C
\end{array}[/tex]
Hint: Let [tex]$u =\tan x$[/tex]
Answer (2)
The integral ∫ t a n x t a n 2 x − 1 s e c 2 x d x simplifies to sec − 1 ∣ tan x ∣ + C using the substitution u = tan x and recognizing the integral form. The final answer is therefore sec − 1 ∣ tan x ∣ + C .
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Substitute u = tan x , so d u = sec 2 x d x .
Rewrite the integral as ∫ u u 2 − 1 d u .
Recognize the integral as sec − 1 ∣ u ∣ + C .
Substitute back to get the final answer: sec − 1 ∣ tan x ∣ + C .
Explanation
Problem Setup and Substitution We are given the integral ∫ tan x tan 2 x − 1 sec 2 x d x and the hint to use the substitution u = tan x . Let's follow this hint.
Applying the Substitution Let u = tan x . Then, the derivative of u with respect to x is d u / d x = sec 2 x , which means d u = sec 2 x d x . Substituting these into the integral, we get ∫ tan x tan 2 x − 1 sec 2 x d x = ∫ u u 2 − 1 d u
Integrating with Inverse Secant Now, we recognize that the integral ∫ u u 2 − 1 d u is the standard integral for the inverse secant function, sec − 1 ∣ u ∣ . Therefore, ∫ u u 2 − 1 d u = sec − 1 ∣ u ∣ + C
Substituting Back Finally, we substitute back u = tan x to express the antiderivative in terms of x :
sec − 1 ∣ u ∣ + C = sec − 1 ∣ tan x ∣ + C
Final Answer Thus, the antiderivative of the given integral is sec − 1 ∣ tan x ∣ + C .
The question asks for the answer in the form [ ? ] − 1 ∣ [ ] x ∣ + C . Comparing this with our result, we have: ∫ tan x tan 2 x − 1 sec 2 x d x = sec − 1 ∣ tan x ∣ + C Therefore, the missing terms are sec and tan .
Examples
Imagine you're designing a robotic arm that needs to move with specific angular velocities. The relationship between the arm's angle and its velocity might involve inverse trigonometric functions. Being able to find antiderivatives of such functions, like the inverse secant, allows you to calculate the arm's exact position over time, ensuring precise and controlled movements. This is crucial for tasks requiring high accuracy, such as surgery or manufacturing.
