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 \(d\)?

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

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

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

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