Indefinite Integration
Rational Function Integration — Parameter Identification
nta_pyq_2024_apr
Grade 12
Question:
If $\displaystyle\int\frac{1}{\sqrt[5]{(x-1)^4(x+3)^6}}\,dx=A\!\left(\dfrac{\alpha x-1}{\beta x+3}\right)^{\!B}+C$, where $C$ is the constant of integration, then the value of $\alpha+\beta+20AB$ is
Step-by-Step Solution
Key Concept: Write integrand as $\frac{1}{(x-1)^{4/5}(x+3)^{6/5}}=\frac{1}{\left(\frac{x-1}{x+3}\right)^{4/5}(x+3)^2}$. Substitute $t=\frac{x-1}{x+3}$, $dt=\frac{4}{(x+3)^2}dx$.
$A=5/4$, $B=1/5$, $\alpha=\beta=1$. $\alpha+\beta+20AB=2+5=7$.
Correct Answer: 7