Fun with exponents

Calculus Level 4

$\Large\int_0^1 \exp({x+e^x+e^{e^x}+e^{e^{e^x}}}) \, dx$

If the above integral can be expressed as

$^Ae - ^B e$

for positive integers $A$ and $B$ , find $A+B$ .

Notations :

$\exp(x)$ denotes the exponential function, $\exp(x) = e^x$ .
$^na$ denotes the Tetration function, $\Large ^n a = \underbrace{a^{a^{a^{\cdot^{\cdot^a}}}}}_{n\text{ number of } a\text{'s}}$ .

The answer is 7.

1 solution

Aareyan Manzoor
Feb 14, 2016

$f(n)=\displaystyle\int_0^1 e^x e^{e^x} .... \underbrace{e^{e^{e^{...}}}}_{\text{n times}} dx$ make the y-sub $y=e^x$ and changing y to x: $f(n)=\displaystyle\int_1^e e^x e^{e^x} .... \underbrace{e^{e^{e^{...}}}}_{\text{n-1 times}} dx$ then by induction we would have $f(n) = \int_{^{n-1}e}^{^n e} dx= ^{n}e-^{n-1}e$ Put n =4 and get the answer.

You got the key to solve this problem. well done!

Nihar Mahajan - 5 years, 4 months ago

a wrong-answered level 4 question :0

Joel Yip - 5 years, 4 months ago

PS it should be 7 as the answer not 9

Aareyan Manzoor - 5 years, 4 months ago

Fun with exponents

The answer is 7.

1 solution

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