Quad Integral?!

Calculus Level 2

∫ ∫ ∫ 1 w z ∫ 1 y 2 x ln ⁡ ( x ) d x d y d w d z \large \int \int \int_{1}^{wz} \int_{1}^{y^2} \sqrt{x}\ln(x)\:dx\:dy\:dw\:dz

Given that the quadruple integral above can be expressed as 1 a ( w z ) b ( ln ⁡ ( w z ) − c − 1 c ) + 1 d w z ( w z − f g ) + C 1 z + C 2 \cfrac{1}{a}(wz)^b\left(\ln(wz)-\cfrac{c-1}{c}\right)+\cfrac{1}{d}wz\left(wz-\cfrac{f}{g}\right)+C_1z+C_2 where C 1 , C 2 C_1, C_2 are constants of integration, a , b , c , d , f , g ∈ N a, b, c, d, f, g \in \N and f , g f, g are co-prime, evaluate a + b + c + d + f + g b \cfrac{a+b+c+d+f+g}{b} .


The answer is 32.4.

James Watson by James Watson , 16, United Kingdom

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Carrying on the integrations we get the final result as

( w z ) 5 75 ( ln ⁡ ( w z ) − 59 60 ) + w z 9 ( w z − 9 4 ) \dfrac {(wz)^5}{75}\left (\ln (wz) -\dfrac {59}{60}\right ) +\dfrac {wz}{9}\left (wz-\dfrac {9}{4}\right )

Hence, a = 75 , b = 5 , c = 60 , d = 9 , f = 9 , g = 4 a=75,b=5,c=60,d=9,f=9,g=4 , and the answer is 75 + 5 + 60 + 9 + 9 + 4 5 = 32.4 \dfrac {75+5+60+9+9+4}{5}=\boxed {32.4} .

0 Helpful 0 Interesting 2 Brilliant 1 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...