Math CentralQuandaries & Queries


Question from Rahul:

How to solve definite integral of a sum. The specific problem is as follows,
Integral of ( 1+ sum of x^k, k=1 to k=n), x=0 to x=b *dx.
The answer is b + sum of b^(k)/k, k=2, to k=n+1. I understand only the integral
of first term. But integral of the sum I do not understand at all.

Hi Rahul,

Let's try this with $n = 3.$ The sum is then

\[ \sum_{k=1}^{k=3} x^k\]

which unpacks to

\[x^1 + x^2 + x^{3}.\]

Thus you have

\[\int_{0}^{b} {1 + x^1 + x^2 + x^3} dx.\]

Evaluate this definite integral and rewrite the answer using summation notation. Can you make it look like the answer given in your question with $n = 3?$

This works because the integral of a sum is the sum of integrals. That is

\[\int_{0}^{b} {1 + x^1 + x^2 + x^3} dx = \int_{0}^{b}{1}\; dx+ \sum_{k=1}^{k=3} {\int_{0}^{b} {x^k}}{dx}. \]

Try to repeat this with $k = 1$ to $k = n.$

Write back if you need more assistance,

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