Integral's'

Calculus Level 2

∫ 0 1 x 1 ∫ 0 2 x 2 2 ∫ 0 3 x 3 3 ∫ 0 4 x 4 4 d x 4 d x 3 d x 2 d x 1 = ? \large \int_{0}^{1} x_1 \int_{0}^{2} {x_2}^2\int_{0}^{3} {x_3}^3\int_{0}^{4} {x_4}^4 \ dx_4 \ dx_3 \ dx_2 \ dx_1 =\, ?


The answer is 5529.6.

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Ashish Menon by Ashish Menon , 20, India

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1 solution

Chew-Seong Cheong
May 19, 2016

∫ 0 1 x ∫ 0 2 x 2 ∫ 0 3 x 3 ∫ 0 4 x 4 d x d x d x d x = x 5 5 ∣ 0 4 ∫ 0 1 x ∫ 0 2 x 2 ∫ 0 3 x 3 d x d x d x = 4 5 5 ⋅ x 4 4 ∣ 0 3 ∫ 0 1 x ∫ 0 2 x 2 d x d x = 4 5 5 ⋅ 3 4 4 ⋅ x 3 3 ∣ 0 2 ∫ 0 1 x d x = 4 5 5 ⋅ 3 4 4 ⋅ 2 3 3 ⋅ x 2 2 ∣ 0 1 = 4 5 5 ⋅ 3 4 4 ⋅ 2 3 3 ⋅ 1 2 = 5529.6 \begin{aligned} \int_0^1 x \int_0^2 x^2 \int_0^3 x^3 \int_0^4 x^4 \ dx \ dx \ dx \ dx & = \frac{x^5}{5} \ \bigg|_0^4 \int_0^1 x \int_0^2 x^2 \int_0^3 x^3 \ dx \ dx \ dx \\ & = \frac{4^5}{5} \cdot \frac{x^4}{4} \ \bigg|_0^3 \int_0^1 x \int_0^2 x^2 \ dx \ dx \\ & = \frac{4^5}{5} \cdot \frac{3^4}{4} \cdot \frac{x^3}{3} \ \bigg|_0^2 \int_0^1 x \ dx \\ & = \frac{4^5}{5} \cdot \frac{3^4}{4} \cdot \frac{2^3}{3} \cdot \frac{x^2}{2} \ \bigg|_0^1 \\ & = \frac{4^5}{5} \cdot \frac{3^4}{4} \cdot \frac{2^3}{3} \cdot \frac{1}{2} \\ & = \boxed{5529.6} \end{aligned}

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Correct! Thanks for the solution!

Ashish Menon - 5 years ago

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Oops I didn't calculated it correctly..

Gaurav Chahar - 5 years ago

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I am sure that you should have given a nice try :) better luck next time. ;)

Ashish Menon - 5 years ago

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