Learn the Basics of Anti-Derivatives: A Comprehensive Guide

[groundpilots]

Anti-Derivatives Please Help!

So I am having trouble understanding where I went wrong here, the question is highlighted, then it clearly states my answer, and the answer the book gave, my work is posted in the second image, care to help??

Homework Statement

The Attempt at a Solution

 

[icystrike]

Hi groundpilots! :)

$$\frac{d}{ds}(a+bs)^{n+1}=(n+1)(b)(a+bs)^{n}$$
<===>
$$(a+bs)^{n+1} + c =\int(n+1)(b)(a+bs)^{n} ds$$

With this can u tell me how u do the antiderivatives ?

If u notice b=5 and n=2 ...

 

[groundpilots]

so I tried to do the antiderivative using the formulas you gave and I still can't seem to get it...ughhhhh
here's what I tried, still not right =(

 

[Mark44]

$$\int (5s + 3)^2~ds$$

Let u = 5s + 3, then du = 5ds

Using this substitution, the integral above becomes:
$$\frac{1}{5}\int (5s + 3)^2~5ds = \frac{1}{5}\int u^2~du$$

Can you take it from here?

BTW, this is an indefinite integral, not an infinite integral.