Indefinite Integration
Matching Integral Form — Maximum of Linear Combination
nta_pyq_2024_apr
Grade 12
Question:
If $\displaystyle\int\frac{dx}{a^2\sin^2 x+b^2\cos^2 x}=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$ is:
$\sqrt{40}$
$\sqrt{41}$
$\sqrt{39}$
$\sqrt{42}$
Step-by-Step Solution
Key Concept: Rewrite integrand: multiply numerator/denominator by $\sec^2 x$, substitute $t=\tan x$ to get $\frac{1}{ab}\tan^{-1}\!\left(\frac{a}{b}\tan x\right)+C$. Comparing: $\frac{1}{ab}=\frac{1}{12}$ and $\frac{a}{b}=3$.
$a=6$, $b=2$. Maximum of $6\sin x+2\cos x=\sqrt{40}$.
Correct Answer: 1