The given equation is:

$\int_{2a}^{9} \left( b\sqrt{x} + \frac{a}{\sqrt{x}} \right) dx = 42$

To solve this, we will first compute the integral of each term separately and then evaluate it within the bounds $2a$ and $9$.

Step 1: Integral of $b\sqrt{x}$

We can rewrite $\sqrt{x}$ as $x^{1/2}$. Therefore, the integral of $b\sqrt{x}$ becomes:

$\int b\sqrt{x} \, dx = b \int x^{1/2} \, dx = b \cdot \frac{2}{3} x^{3/2}$

Step 2: Integral of $\frac{a}{\sqrt{x}}$

We can rewrite $\frac{a}{\sqrt{x}}$ as $a x^{-1/2}$. Therefore, the integral of $\frac{a}{\sqrt{x}}$ becomes:

$\int \frac{a}{\sqrt{x}} \, dx = a \int x^{-1/2} \, dx = a \cdot 2 x^{1/2}$

Step 3: Combine the results

Now, the integral becomes:

$\int \left( b\sqrt{x} + \frac{a}{\sqrt{x}} \right) dx = \left( b \cdot \frac{2}{3} x^{3/2} + a \cdot 2 x^{1/2} \right)$

Step 4: Evaluate at bounds $2a$ and $9$

We now evaluate the combined result at the upper and lower bounds and set the result equal to 42:

$\left[ b \cdot \frac{2}{3} x^{3/2} + a \cdot 2 x^{1/2} \right]_{2a}^{9} = 42$

Substitute $x = 9$ and $x = 2a$ into the expression, and then solve for $a$ and $b$.

Let's evaluate these integrals for the given limits. I'll compute the result next.The evaluated definite integral is:

$\left( 6a + 18b - \frac{4\sqrt{2} a^{3/2} b}{3} - 2\sqrt{2} a^{3/2} \right) = 42$

This is the equation that needs to be solved for $a$ and $b$.

To proceed, you would typically need additional information to either solve for $a$ or $b$ (such as initial conditions or other constraints). If that information is available, we can substitute it into the equation and solve.

Would you like to proceed with solving for either variable, or do you have additional information that could help narrow it down?

Relative questions:

1. Would you like a more detailed explanation of each step of the integration?
2. Should we assume any specific values for $a$ or $b$ to simplify the equation?
3. Do you want to solve for $a$ or $b$ explicitly?
4. Are there any other equations or conditions related to this problem?
5. Do you need a graph of the integral or any related function?

Tip:

Always check the bounds of an integral to ensure they align with the context of the problem; incorrect limits can lead to errors.