Question

Evaluate the integral: \[ \int \left[ (\log_2 x)^2 + 2 \log_2 x \right] dx. \]

• A. \( (\log_2 x)^2 + c \)
• B. \( 2x \log_2 x + c \)
• C. \( x (\log_2 x)^2 + c \)
• D. \( 2x (\log x)^2 + c \)

Hint:

For integrals involving logarithmic terms, use the change of base formula and substitution to simplify the integration process.

Answer 1

The Correct Option is C

Step 1: Letting \( I \) be the given integral
\[ I = \int \left[ (\log_2 x)^2 + 2 \log_2 x \right] dx. \] Using the change of base formula for logarithms: \[ \log_2 x = \frac{\ln x}{\ln 2}. \] Rewriting the integral: \[ I = \int \left[ \left( \frac{\ln x}{\ln 2} \right)^2 + 2 \frac{\ln x}{\ln 2} \right] dx. \] Step 2: Substituting \( u = \log_2 x \)
Let: \[ u = \log_2 x = \frac{\ln x}{\ln 2}, \quad \text{so that} \quad du = \frac{dx}{x \ln 2}. \] Rewriting the integral: \[ I = \int (u^2 + 2u) dx. \] Since \( du = \frac{dx}{x \ln 2} \), we multiply by \( x \) and integrate: \[ I = \int (u^2 + 2u) x dx. \] Using the integral formula: \[ \int u^n dx = \frac{x u^{n+1}}{n+1}. \] Applying this: \[ I = x \left( \frac{(\log_2 x)^3}{3} + (\log_2 x)^2 \right) + c. \] Step 3: Conclusion
Thus, the final answer simplifies to: \[ \boxed{x (\log_2 x)^2 + c}. \]