Indefinite Integration
Power-Log Integral — Bonus Question
nta_pyq_2023_apr
Grade 12
Question:
The integral $\displaystyle\int\!\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right]\log_2 x\,dx$ is equal to
\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x+C
\left(\frac{x}{2}\right)^x-\left(\frac{2}{x}\right)^x+C
\left(\frac{x}{2}\right)^x\log_2(x)+C
\left(\frac{x}{2}\right)^x\log_2\!\left(\frac{2}{x}\right)+C
Step-by-Step Solution
Key Concept: Let $t=(\frac{x}{2})^x$. Then $\ln t=x\ln(x/2)$ and $dt=t(\ln(x/2)+1)dx$. Similarly handle $(\frac{2}{x})^x$.
$\int\left[(\frac{x}{2})^x+(\frac{2}{x})^x\right]\log_2 x\,dx=(\frac{x}{2})^x+(\frac{2}{x})^x+C$.
Correct Answer: 1