Let the slope of the line $45 x+5 y+3=0$ be $27 r_1+\frac{9 r_2}{2}$ for some $r_1, r_2 \in \mathbb{R}$. Then $$\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)$$ is equal to _________.
Answer (integer)
12
Solution
<p>According to the question,</p>
<p>$$\begin{aligned}
& 27 r_1+\frac{9 r_2}{2}=-9 \\
& \lim _\limits{x \rightarrow 3} \frac{\int_\limits3^x 8 t^2 d t}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} \\
& =\lim _\limits{x \rightarrow 3} \frac{8 x^2}{\frac{3 r_2^2}{2}-2 r_2 x-3 r_1 x^2-3} \text { (using LH' Rule) } \\
& =\frac{72}{\frac{3 r_2}{2}-6 r_2-27 r_1-3} \\
& =\frac{72}{-\frac{9 r_2}{2}-27 r_1-3} \\
& =\frac{72}{9-3}=12
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Definite Integration · Topic: Properties of Definite Integrals
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