We are tasked with evaluating the following definite integral:

$\int_1^4 3 \ln(9 + 3s) \, ds$

Let's solve this step by step:

Step 1: Substitution

To simplify the integral, we use substitution. Let: $u = 9 + 3s$ Thus, the derivative of $u$ with respect to $s$ is: $du = 3 \, ds \quad \text{or} \quad ds = \frac{du}{3}$

Also, update the limits of integration according to the substitution:

• When $s = 1$, $u = 9 + 3(1) = 12$,
• When $s = 4$, $u = 9 + 3(4) = 21$.

Step 2: Update the integral

Substitute into the integral: $\int_{1}^{4} 3 \ln(9 + 3s) \, ds = \int_{12}^{21} 3 \ln(u) \cdot \frac{du}{3}$

The $3$'s cancel out, so we are left with: $\int_{12}^{21} \ln(u) \, du$

Step 3: Integration by parts

We need to integrate $\ln(u)$. This is a standard integral that can be solved using integration by parts. Let: $v = \ln(u) \quad \text{and} \quad dw = du$ Then, $dv = \frac{1}{u} \, du \quad \text{and} \quad w = u$

Now, apply the integration by parts formula: $\int v \, dw = v \cdot w - \int w \, dv$ Substituting the terms: $\int \ln(u) \, du = u \ln(u) - \int u \cdot \frac{1}{u} \, du = u \ln(u) - u + C$

Step 4: Evaluate the integral

Now, evaluate this at the limits $u = 21$ and $u = 12$: $\left[ u \ln(u) - u \right]_{12}^{21} = \left( 21 \ln(21) - 21 \right) - \left( 12 \ln(12) - 12 \right)$

Step 5: Approximate the result

Let's compute the numerical values of this expression.

$21 \ln(21) \approx 21 \times 3.0445 = 63.9345$ $12 \ln(12) \approx 12 \times 2.4849 = 29.8188$ Now substitute: $(63.9345 - 21) - (29.8188 - 12) = 42.9345 - 17.8188 = 25.1157$

Thus, the value of the integral is approximately: $\boxed{25.12}$

Would you like further clarification or a more detailed breakdown of any step?

Here are 5 related questions for further practice:

1. How would the integral change if the bounds were from 0 to 3 instead of 1 to 4?
2. Can you compute the integral $\int \ln(3x + 5) \, dx$?
3. How does the substitution method simplify complex integrals?
4. What are the main steps of integration by parts, and when should it be used?
5. What if the function inside the logarithm was quadratic instead of linear? How would that affect the solution?

Tip: Always check if substitution can simplify the integration before using more advanced techniques like integration by parts.