This Many Logs Inside An Integral Makes Me Sick!

Calculus Level 4

$\large \int_e^{e^{512}} \dfrac{dx}{x \ln x + x \ln x ( \ln (\ln x))^2 }$

If the integral above is equal to $\arctan\left(\ln A^{B^C} \right)$ , where $A,B$ and $C$ are prime numbers, find $A+B+C$ .

The answer is 7.

1 solution

Rishabh Jain
Feb 4, 2016

Just put $\ln(\ln x)=t$ such that dt= $\dfrac{dx}{x \ln x}$ and integral simplifies to: $\Large \int_0^{\ln 512} \dfrac{dt}{1+t^2}$ $\Large = (\arctan t)|_{\small{0}}^{\small{\ln 512}}$ $\Large =\arctan (\ln 512) = \arctan (\ln 2^{3^2})$ $\Large 2+3+2=\huge\boxed{\color{#007fff}{ 7}}$

Lovely man!

Adarsh Kumar - 5 years, 4 months ago

$\Large\color{#302B94}{\mathcal{Thanks!!}}$

Rishabh Jain - 5 years, 4 months ago

same method!

awesome solution

Hamza A - 5 years, 4 months ago

$\Large\color{#456461}{\mathbb{Thanks!!}}$

Rishabh Jain - 5 years, 4 months ago

just one question

how do you write thanks like that (using LaTeX?)

it looks cool

Thanks

Hamza A - 5 years, 4 months ago

@Hamza A – Enclose text in \mathbb{text}. Like I used \mathbb{Thanks!!}

Rishabh Jain - 5 years, 4 months ago

This Many Logs Inside An Integral Makes Me Sick!

The answer is 7.

1 solution

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