Too many variables?

Calculus Level 3

Given that

$\int_{B}^{2} \left (2 + \frac {A} {5x-A} \right ) \,dx = A - B + \frac {3} {5} \ln \frac {C}{D}$

where $C$ and $D$ are coprime positive integers, and

$5x - A > 0$

Submit the value of $C + D$ .

6

7

8

9

1 solution

Ethan Mandelez
May 24, 2021

Computing this definite integral we get

$(4 + \frac {A} {5} \ln (10-A) ) - (2B + \frac {A} {5} \ln (5B-A) )$

Since it is given that $5x - A > 0$ , there is no need for a modulus sign.

Simplifying we get

$(4 - 2B) + \frac {A} {5} \ln \frac {(10-A)}{(5B-A)}$

Since this is also equal to

$A - B + \frac {3} {5} \ln \frac {C}{D}$

We can conclude that

$A - B = 4 - 2B$

$\frac {3} {5} = \frac {A} {5}$

From the second equation $A = 3$ and substituting $A$ into the first equation we get $B = 1$ .

Therefore

$\int_{1}^{2} (2 + \frac {3} {5x-3}) \,dx = 2 + \frac {3} {5} \ln \frac {7}{2}$

where $C = 10 - A = 7$ and $D = 5B - A = 2$

The answer is $9$ .