# 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
4 years, 11 months ago

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

- 4 years, 11 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.

- 4 years, 11 months ago

Here is what I think:

Rewrite the integrand as

$$(1 + x + x^2 + x^3 + ...) - x$$

then we can rewrite it as

$$\frac{1}{1 - x} - x$$

Thus the integrand becomes

$$\int^{3}_2{\frac{1}{1 - x} - x}$$

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

- 4 years, 11 months ago

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

- 4 years, 11 months ago