Question

If P is the function defined by P(x)=a1+2a2x⋅⋅⋅ +(n+1)an+1xn write down an expression for \int^b_aP(x)dx Justify your answer

Answer 1

$\displaystyle P(x)=a_1+2a_2x +...+(n+1)a_{n+1}x^n\\ P(x) \,\textsf{can generally be written as,}\\ P(x) = \sum_{k = 1}^{n + 1} k a_k x^{k - 1}\\ \int_a^b P(x) \, \mathrm{d}x = \int_a^b \sum_{k = 1}^{n + 1} k a_k x^{k - 1}\, \mathrm{d}x\\ \textsf{Interchanging summation for integration, we have}\\ \int_a^b P(x) \, \mathrm{d}x = \sum_{k = 1}^{n + 1} k a_k \int_a^b x^{k - 1}\, \mathrm{d}x\\ \int_a^b P(x) \, \mathrm{d}x = \sum_{k = 1}^{n + 1} \cancel{k} a_k \frac{x^{k}}{\cancel{k}}\vert_a^b + C\\ \therefore\int_a^b P(x) \, \mathrm{d}x = \sum_{k = 1}^{n + 1} a_k (b^k - a^k)+ C\\$