   SEARCH HOME Math Central Quandaries & 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,
Penny      Math Central is supported by the University of Regina and the Imperial Oil Foundation.