Challenge Level

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 $d$?**

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

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

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

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