5 day streak problem original!

Calculus Level pending

$\large \int_0^1 \int_0^2 \int_0^3 \left( x \ \ln(xyz) \right) \, dx \, dy \, dz$

If the expression above equals to $a \ln(b) - c$ for integers $a,b,c$ with $b$ square free, find the value of $abc$ .

1 solution

Aman Rajput
Jun 17, 2015

Since all the limits are constant and independent,therefore we can separately integrate them. Integral can be rewrite in the form of :

$\displaystyle\int_0^1\int_0^2\int_0^3 x\log(x) + x\log(y) + x\log(z) dx dy dz$

$\displaystyle\int_0^1\int_0^2 \frac{9}{4}(\log(9)-1) + \frac{9}{2}\log(y) + \frac{9}{2}\log(z) dy dz$

$\displaystyle\int_0^1 \frac{9}{2}(\log(9)-1) + \frac{9}{2}(\log(4)-2) + 9\log(z) dz$

$\displaystyle\frac{9}{2}(\log(9)-1) + \frac{9}{2}(\log(4)-2) - 9$

$\displaystyle\frac{9}{2}(\log(36)-5)$

$\displaystyle9\log(6) - \frac{45}{2}$

$abc=1215$

Wow.I was expecting some people to integrate with parts but brute force.Well done!!!

Timothy Wan - 5 years, 12 months ago

Thanks friend ! :)

Aman Rajput - 5 years, 12 months ago

By the way its $log_e$ ,not $log_{10}$

Timothy Wan - 5 years, 12 months ago

@Timothy Wan – by default $\log$ Means natural logarithm

Aman Rajput - 5 years, 11 months ago