# Integral chasing

Can you find the missing constants from these not-quite-so-obvious definite integrals?

## Problem

Below is a selection of integrals, some of which you can't do directly and some that require the use of other mathematical skills.**Can you find the missing positive numbers $a$ to $d$?**

$$\text{(1)} \ \int_a^5 10x+3 \ \text{d}x = 114$$

$$\text{(2)} \ \int_{2a}^9 b\sqrt{x}+\dfrac{a}{\sqrt{x}}\ \text{d}x=42$$

$$\text{(3)} \ \int_{\frac{1}{2}}^1 \dfrac{1}{x^5}-\dfrac{1}{x^2} \ \text{d}x=\dfrac{c+1}{4}$$

$$\text{(4)} \ \int^{c+2}_6 x^{\frac{b}{a}}\left(\sqrt{x}-\dfrac{1}{\sqrt{x}}\right) \ \text{d}x=ab^ad^a$$

**This is an Underground Mathematics resource.**

*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*

*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*

## Getting Started

Once you have found the missing positive numbers $a$ to $d$, can you use them in the statement below to test your values?

The area formed between the $x$-axis and the lines $x=b$ and $x=d$, and the curve $y=(x-2a)(x+1)$ is $\dfrac{cd}{a(a+b)}$.

The area formed between the $x$-axis and the lines $x=b$ and $x=d$, and the curve $y=(x-2a)(x+1)$ is $\dfrac{cd}{a(a+b)}$.

*Remember, when we are looking at area, what must we check about the curve between the lines $x=b$ and $x=d$ when it is plotted?*## Student Solutions

Hayley from Kimberley STEM College has managed to find the values of a,b,c and d which are:

$a=2$

$b=3$

$c=10$

$d=5$

Here is her working.