Definite Integration
Area bounded by implicit curve and lines
MJAT_TS8_P1
Grade 12
Question:
If $\displaystyle\int\left(\frac{x^6-1}{x^3+5x^2+30x^8}\right)\log_e x\right)^{7/8}dx = \frac{15}{4}(f(x))^{18}+C$ and $f(1)=11$, then the area bounded by $y=f(x)$, $x=\frac{1}{2}$, $x=1$ and $y=11$ is $p+q\ln 2$ where $p,q\in\mathbb{N}$. The value of $p+q$ is:
Step-by-Step Solution
Key Concept: Find $f(x)$ from the integral. The integrand suggests $f(x)=6x^{-5}+5x^{-6}+30\ln x$ (from the solution structure). Then compute the area $A=\int_{1/2}^1|f(x)-11|dx=p+q\ln 2$.
$p+q=\mathbf{48}$.
Correct Answer: 48