Integration#2

Calculus Level pending

In the diagram above, $l_1, l_2$ are straight lines, whose slopes are -1 and 0, respectively.

Denote the slope of $OA$ as $m$ and $k = \dfrac{\overline{AB} \cdot \overline{AC}}{\Delta OBC}.$

$k$ can be expressed as $\dfrac{\sqrt{ f(m)}}{m},$ where $f(m)$ is a polynomial with respect to $m$ .

Given that $\displaystyle \int _{\frac12}^2 \frac{\sqrt{f(x)}}x \, dx = \ln{p}-q$ , find the value of $(p-1.5)q$ .

( $p-1.5, q$ are expressed as $\sqrt{ \dfrac{a}{b}}$ where $a,b$ are positive integers.)

The answer is 1.875.

1 solution

Inquisitor Math
May 3, 2021

Let point $B(a,2^a)$ , then $A(2^a,a)$ . Note that $m = a•2^{-a}$ . Express $k$ using $a$ . By dividing the denominator and numerator of $k$ by $2^{2a}$ , this yields $f(m) = 4(1-m)^2(1+m^2)$ . Thus the final results are $p = \dfrac{3+ \sqrt5}{2}, q = \dfrac{3\sqrt5}{4}$

Note: $m,k$ are originally constants, about 0.303436,4.797873 respectively.

Integration#2

The answer is 1.875.

1 solution

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