Given that $y = 5x^3 + 7x + 3$, find (a) $\frac{dy}{dx}$ - Edexcel - A-Level Maths Pure - Question 6 - 2005 - Paper 2

Now, we find the second derivative by differentiating the first derivative:

1. The derivative of $15x^2$ is $30x$.
2. The derivative of $7$ is $0$.

Therefore, the second derivative is:

$\frac{d^2y}{dx^2} = 30x.$

We will calculate the integral term by term:

1. The integral of $1$ is $x$.
2. The integral of $3\sqrt{x}$ is $3 \cdot \frac{2}{3} x^{3/2} = 2x^{3/2}$.
3. The integral of $-\frac{1}{x}$ is $-\ln|x|$.

Combining these results, the final answer is:

$\int\left[1 + 3\sqrt{x} - \frac{1}{x}\right]dx = x + 2x^{3/2} - \ln|x| + C,$ where $C$ is the constant of integration.