Establish and use the formula ∫ 1 𝑥 𝑑𝑥 = ln|𝑥| + 𝑐 for 𝑥 ≠ 0 and ∫ 𝑓′(𝑥) 𝑓(𝑥) 𝑑𝑥 = ln|𝑓(𝑥)| + 𝑐 for 𝑓(𝑥) ≠ 0.

Consider the function \( f(x) = 3x^2 - 7 \). (a) Determine \( \frac{d}{dx}[\ln|f(x)|] \), simplifying your answer. (b) Hence find \( \int \frac{6x}{3x^2 - 7} \, dx \).

Consider the shaded region bounded by the curve $y = \frac{3}{2x+1}$, the $x$-axis, and the vertical lines $x = 1$ and $x = 4$, as shown in the diagram below. What is the area of the shaded region?

Consider the function g(x) = 3x² + 4x − 7. (a) Show that ∫ (6x + 4)/(3x² + 4x − 7) dx = ln|3x² + 4x − 7| + c for 3x² + 4x − 7 ≠ 0. (2 marks) (b) Hence, evaluate ∫₂³ (6x + 4)/(3x² + 4x − 7) dx, expressing your answer in exact form. (3 marks)

Determine \(\displaystyle\int \frac{6x^2}{2x^3 + 5} \, dx\).