Question

Given \[ f(x)=\int \frac{dx}{x^{2/3}+2\sqrt{x}} \quad \text{and} \quad f(0)=-26+24\ln 2. \] If \(f(1)=A+B\ln 3\), then find \((A+B)\).

• A. \(10\)
• B. \(11\)
• C. \(-11\)
• D. \(-10\)

Hint:

When integrals involve mixed powers of \(x\), try a substitution that converts all powers into simple polynomials.

Answer 1

The Correct Option is C

To solve the given problem, we need to evaluate the function \( f(x) \) given by the integral:

\(f(x) = \int \frac{dx}{x^{2/3}+2\sqrt{x}}\)

and use the initial condition: \(f(0) = -26 + 24\ln 2\).

We also have the condition: \(f(1) = A + B\ln 3\), and we need to find the value of \((A+B)\).

1. First, simplify the given expression in the integrand:
2. Find \(dx\) in terms of \(dt\):
3. Substitute these into the integral:
4. Solve the integral with limits:
5. Apply the initial condition:
6. Find \(f(1)\): Resulting in \(A + B\ln 3\):
7. Finally, calculate \((A + B)\):

Thus, the answer is \(-11\).