Integral Calculus
Integral Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
If $f(x)$ is an odd function, then:
\phi(x) is also an odd function
\phi(x) is an even function
\phi(x) is neither an even nor an odd function
For \phi(x) to be an even function, it must satisfy \int_0^a f(x) dx = 0
Step-by-Step Solution
Key Concept: Converting to a rational function via $t = \tan x$ enables the use of partial fractions.
Rewrite the integral as $I = \int \frac{\sec^2 x}{\tan^2 x - 4} dx$. Substitute $\tan x = t$ to get $I = \int \frac{dt}{t^2 - 4}$. Using partial fractions: $\frac{1}{t^2-4} = \frac{1}{4}\left(\frac{1}{t-2} - \frac{1}{t+2}\right)$, yielding $I = \frac{1}{4}\ln\left|\frac{\tan x - 2}{\tan x + 2}\right| + C$.
Correct Answer: 1,2