$\int \tan ^2 x dx=$
A. $\sec ^2 x+c$
B. $2 \tan x+c$
C. $\tan x-x+c$
D. $2 \tan x-x+c$
Answer (1)
Use the trigonometric identity tan 2 x = sec 2 x − 1 .
Rewrite the integral as ∫ ( sec 2 x − 1 ) d x .
Split the integral into two separate integrals: ∫ sec 2 x d x − ∫ 1 d x .
Evaluate the integrals to get tan x − x + C . Thus, the answer is tan x − x + c .
Explanation
Problem Analysis We are asked to find the indefinite integral of tan 2 x .
Using Trigonometric Identity We can use the trigonometric identity tan 2 x = sec 2 x − 1 to rewrite the integral.
Rewriting the Integral So, the integral becomes ∫ tan 2 x d x = ∫ ( sec 2 x − 1 ) d x .
Splitting the Integral We can split the integral into two separate integrals: ∫ sec 2 x d x − ∫ 1 d x .
Evaluating the Integrals The integral of sec 2 x is tan x , and the integral of 1 is x . Therefore, we have tan x − x + C , where C is the constant of integration.
Comparing with Options Comparing our result with the given options, we see that option C matches our result.
Examples
In physics, when analyzing the motion of a damped harmonic oscillator, you might encounter integrals involving trigonometric functions. Evaluating ∫ tan 2 x d x can be useful in determining the energy dissipation or the average kinetic energy of the system over a certain period. Understanding how to integrate trigonometric functions is crucial for solving problems related to oscillations and waves.
