Is it partial?

Calculus Level 3

∫ n n + 1 \displaystyle\int_n^{n+1} f ( x ) f(x) d x dx = n 2 + n n^2+n

For all n n belongs to Integers then evaluate

\quad ∫ − 3 3 \displaystyle\int_{-3}^3 f ( x ) f(x) d x dx


The answer is 16.

Parth Lohomi by Parth Lohomi , 20, India

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1 solution

∫ − 3 − 3 f ( x ) d x = ∫ − 3 − 2 f ( x ) d x + ∫ − 2 − 1 f ( x ) d x + ∫ − 1 0 f ( x ) d x + ∫ 0 1 f ( x ) d x + ∫ 1 2 f ( x ) d x + ∫ 2 3 f ( x ) d x = ( − 3 ) 2 − 3 + ( − 2 ) 2 − 2 + ( − 1 ) 2 − 1 + 0 + 0 + 1 2 + 1 + 2 2 + 2 = 16 \displaystyle \int_{-3}^{-3} f(x)dx \\ =\int_{-3}^{-2} f(x)dx +\int_{-2}^{-1} f(x)dx +\int_{-1}^{0} f(x)dx \\+ \int_{0}^{1} f(x)dx +\int_{1}^{2} f(x)dx + \int_{2}^{3} f(x)dx \\=(-3)^2-3~~+~~(-2)^2-2~~+(-1)^2-1~~+0+0~~+1^2+1~~+2^2+2\\ =\boxed{ 16 }

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Correct !!!!

Parth Lohomi - 6 years, 5 months ago

or could we find f(x) using lebinitz's rule

Ilaya S - 6 years, 5 months ago

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