Integral of the year

Calculus Level 5

lim ⁡ n → ∞ n ∫ 0 1 x n 2014 + x n d x = a \lim_{n\to\infty}n\int_0^1\frac{x^n}{2014 +x^n}dx= a

find [ 1 0 7 a ] [10^{7}a]

[.] - greatest integer function


The answer is 4964.

U Z by U Z , 24, Guyana

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1 solution

Set y = x n y=x^n and use dominated convergence theorem , we will obtain a = ∫ 0 1 d y 2014 + y = ln ⁡ ( 2015 2014 ) a=\int_0^1 \frac{dy}{2014+y}=\ln\left(\frac{2015}{2014}\right)

5 Helpful 0 Interesting 0 Brilliant 0 Confused

You are just 14 year old and have a vast knowledge about integrals , can you suggest me a book to improve my skills @Anastasiya Romanova

sandeep Rathod - 6 years, 6 months ago

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