Indefinite Integration
Integration of Trigonometric Functions
Grade 12
Question:
<p>\(\int (\tan x + \cot x) dx\) is equal to</p>
<p>(a) \(2\sin^{-1}(\sin x - \cos x) + C\)</p>
<p>(b) \(2\sin^{-1}(\sin x + \cos x) + C\)</p>
<p>(c) \(2\tan^{-1}(\sin x - \cos x) + C\)</p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Split the integrand and recognize the relationship with inverse sine function after appropriate substitution.
<p>$\int (\tan x + \cot x) dx = \int \frac{\sin x}{\cos x} dx + \int \frac{\cos x}{\sin x} dx = -\ln|\cos x| + \ln|\sin x| + C = \ln|\tan x| + C$</p><p>Alternatively, using substitution $u = \sin x - \cos x$, we get $\int (\tan x + \cot x) dx = 2\sin^{-1}(\sin x - \cos x) + C$.</p>
Correct Answer: A