Fluency exercise

## Problem

Below is a selection of integrals, some of which you cannot do without performing others first and some that require the use of other mathematical skills.

Can you find the missing positive numbers $a$ to $d$?

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

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

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

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