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 \(d\)?

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

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

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

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

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 by the \(x\)-axis, 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?