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Problem Summary:

We are given the definite integral: $I = \int_{2}^{7^{1/3}} \frac{2x}{\sqrt{3x+2}} dx$ The task involves:

1. Selecting an appropriate substitution from a list.
2. Computing the integral by changing the variable, adjusting the integration limits, and evaluating the result.

Step-by-Step Solution:

1. Choose Substitution:

From the structure of the integral, notice the expression $\sqrt{3x+2}$ in the denominator. A substitution related to $3x+2$ simplifies the integral. The substitution is: $x = \frac{1}{3} t^2 - \frac{2}{3}$ where:

• $dx = \frac{2}{3} t \, dt$
• $3x+2 = t^2$.

2. Change the Integration Limits:

• When $x = 2$: $3(2) + 2 = t^2 \implies t = \sqrt{8}.$

• When $x = 7^{1/3}$: $3\left(7^{1/3}\right) + 2 = t^2 \implies t = \sqrt{3 \cdot 7^{1/3} + 2}.$

Thus, the new limits are: $A = \sqrt{8}, \quad B = \sqrt{3 \cdot 7^{1/3} + 2}.$

3. Substitute into the Integral:

After substitution, the integral becomes: $I = \int_{A}^{B} \frac{2 \cdot \frac{1}{3}(t^2 - \frac{2}{3})}{t \cdot t} \cdot \frac{2}{3} t \, dt.$ Simplify: $I = \int_{A}^{B} \frac{\frac{4}{9} (t^2 - \frac{2}{3})}{t^2} \, dt.$

Separate the terms: $I = \frac{4}{9} \int_{A}^{B} \left(1 - \frac{2}{3t^2}\right) \, dt.$

Split the integral: $I = \frac{4}{9} \left[\int_{A}^{B} 1 \, dt - \frac{2}{3} \int_{A}^{B} \frac{1}{t^2} \, dt\right].$

4. Evaluate the Integral:

• First term: $\int_{A}^{B} 1 \, dt = [t]_{A}^{B} = B - A.$

• Second term: $\int_{A}^{B} \frac{1}{t^2} \, dt = \left[-\frac{1}{t}\right]_{A}^{B} = -\frac{1}{B} + \frac{1}{A}.$

Substitute back: $I = \frac{4}{9} \left[(B - A) - \frac{2}{3} \left(-\frac{1}{B} + \frac{1}{A}\right)\right].$

5. Numerical Evaluation:

1. Calculate $A = \sqrt{8} \approx 2.828$.
2. Calculate $B = \sqrt{3 \cdot 7^{1/3} + 2}$:
   • Approximate $7^{1/3} \approx 1.913$.
   • $B = \sqrt{3 \cdot 1.913 + 2} = \sqrt{7.739} \approx 2.779$.

Substitute $A$ and $B$ into the expression for $I$: $I = \frac{4}{9} \left[(2.779 - 2.828) - \frac{2}{3} \left(-\frac{1}{2.779} + \frac{1}{2.828}\right)\right].$

Compute step-by-step:

• $B - A = 2.779 - 2.828 = -0.049$.
• $-\frac{1}{B} + \frac{1}{A} = -\frac{1}{2.779} + \frac{1}{2.828} \approx -0.360 + 0.354 = -0.006$.
• Multiply by $-\frac{2}{3}$: $-\frac{2}{3} \cdot -0.006 \approx 0.004$.
• Combine: $-0.049 + 0.004 = -0.045$.
• Multiply by $\frac{4}{9}$: $\frac{4}{9} \cdot -0.045 \approx -0.02$.

Final result: $I \approx -0.02.$

6. Final Answer:

The value of the definite integral to 2 decimal places is: $I \approx -0.02.$

Would you like further clarification or detailed steps?

Related Questions:

1. What other substitutions could simplify integrals involving radicals?
2. How does the choice of substitution affect the limits of integration?
3. What numerical methods can approximate definite integrals if manual computation is too complex?
4. Can this integral be solved using symbolic computation software?
5. How is the substitution method applied in solving improper integrals?

Tip:

Always simplify your substitution by aligning it with the structure of the given integral (e.g., matching square roots or denominators).