Integral Calculus-2
Integral Calculus-2
Allen Star Batch
Grade 12
Question:
If $m, n$ are even integers and $p, q \in \mathbb{R}$, then $\int_{p+ma}^{q+na} g(t)dt$ is equal to:
$\int_p^q g(x)dx$
$(n-m)\int_0^a g(x)dx$
$\int_p^q g(x)dx + (n-m)\int_0^a g(2x)dx$
$\int_p^q g(x)dx + (n-m)\int_0^a g(x)dx$
Step-by-Step Solution
Key Concept: Periodic functions allow splitting integrals over translated intervals into sums of complete periods and partial intervals.
The integral $\int_{p+ma}^{q+ma} g(t)dt$ can be split into $m$ complete periods plus the base integral: $\int_{p+ma}^{q+ma} g(t)dt = -m\int_0^a g(t)dt + \int_p^a g(t)dt + n\int_0^p g(t)dt - \int_0^a g(t)dt + (n-m)\int_0^a g(t)dt$ when $b, k, c$ are in arithmetic progression with appropriate indexing of periods.
Correct Answer: 4