If [ 1a^2 ^2 x+b^2 ^2 x ~d x=(1)/(12) ^-1(3 x)+] constant, then the maximum value of [a x+…

Question

If $$\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+$$ constant, then the maximum value of $$\mathrm{a} \sin x+\mathrm{b} \cos x$$, is :

Accepted Answer

$$\begin{aligned}
& \int \frac{1}{a^2 \sin ^2 x+b^2 \cos ^2 x} d x=\frac{1}{12} \tan ^{-1}(3 \tan x)+c \
& I=\int \frac{\sec ^2 x}{b^2+a^2 \tan ^2 x} d x \
& \tan x=t \
& \Rightarrow \sec ^2 x d x=d t \
& I=\int \frac{d t}{b^2+a^2 t^2} \
& =\frac{1}{b a} \tan ^{-1}\left(\frac{a t}{b}\right)+c \
& I=\frac{1}{a b} \tan ^{-1}\left(\frac{a}{b} \tan x\right)+c \
& \Rightarrow a b=12 \text { and } \frac{a}{b}=3\
& \Rightarrow a^2=36 \text { and } b^2=4\
& \Rightarrow \text { Maximum value of } a \sin x+b \cos x \text { is } \sqrt{a^2+b^2}
\end{aligned}$$

$$\quad=\sqrt{36+4}$$

$$\quad=\sqrt{40}$$

Solutions · 1