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1. Rewrite the integrand using fractional indices to prepare for integration

$\int \left( 3(2+7x)^{-\frac{1}{2}} + \sin 5x \right) \, dx$

2. Integrate the power term with a linear argument

$\int 3(2+7x)^{-\frac{1}{2}} \, dx = \frac{3(2+7x)^{\frac{1}{2}}}{\frac{1}{2} \times 7} = \frac{6}{7}\sqrt{2+7x}$

3. Integrate the trigonometric term and combine with the constant of integration

$\int \sin 5x \, dx = -\frac{1}{5}\cos 5x$

Combining the results:
$= \frac{6}{7}\sqrt{2+7x} - \frac{1}{5}\cos 5x + C$

1. Integrate the given derivatives to find the general forms of the functions

Since $f'(x) = 2x$, integrating gives $f(x) = x^2 + C_1$.
Similarly, since $g'(x) = 2x$, integrating gives $g(x) = x^2 + C_2$.

2. Explain why the functions are not necessarily equal

The constants of integration $C_1$ and $C_2$ are arbitrary and can be different. Therefore, $f(x)$ and $g(x)$ are not necessarily the same function; they can differ by a constant.

1. State the property of the exponential function

For all real values of $x$, $e^x > 0$.

2. Deduce the restriction on V

Since $e^x > 0$, it follows that $3e^x > 0$.
Therefore, $V = 20 - 3e^x < 20$.
Hence, $V$ can never be equal to 20.