Definite Integral

If f(x)=1+x^2+x^3+..., determine the integral of f(x)dx from 2 to 3.

Note by Mark Anthony Briones
7 years, 10 months ago

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Comments

What is f(x)f(x)? As currently stated, f(x)=1+x2+x3+…f(x) = 1 + x^2 + x^3 + \ldots which doesn't lend itself to a nice pattern to continue the ellipsis.

Ivan Koswara - 7 years, 10 months ago

I'm awful in calculus, but I think you can separate the expression on infinite integrals, according to the property that says: the integral of a sum is the sum of the integrals. However, I can't see a clear way to solve it, indeed I think there is a solution.

Leonardo Cidrão - 7 years, 10 months ago

Here is what I think:

Rewrite the integrand as

(1+x+x2+x3+...)−x(1 + x + x^2 + x^3 + ...) - x

then we can rewrite it as

11−x−x\frac{1}{1 - x} - x

Thus the integrand becomes

∫2311−x−x\int^{3}_2{\frac{1}{1 - x} - x}

Can you take it from there? From here it's trivial.

Joshua Siktar - 7 years, 10 months ago

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This approach is completely invalid. ∑i=0∞xi=11−x\displaystyle \sum_{i=0}^{\infty} x^i = \frac{1}{1-x} only when ∣x∣<1|x| < 1. Thus this definite integral is actually infinity.

Bob Krueger - 7 years, 10 months ago
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