Indefinite Integration
Radical Substitution — Finding α⁴
nta_pyq_2023_apr
Grade 12
Question:
Let $I(x)=\displaystyle\int\sqrt{\dfrac{x+7}{x}}\,dx$ and $I(9)=12+7\ln 7$. If $I(1)=\alpha+7\ln(1+2\sqrt{2})$, then $\alpha^4$ is equal to _____.
Step-by-Step Solution
Key Concept: Substitute $\frac{x+7}{x}=t^2\Rightarrow dx=\frac{-14t}{(t^2-1)^2}dt$. After integration, use the boundary condition $I(9)=12+7\ln7$ to determine $C$.
$\alpha=2\sqrt{2}$. $\alpha^4=64$.
Correct Answer: 64