Definite Integration
IBP with boundary conditions
MJAT_TS6_P2
Grade 12
Question:
**Paragraph II:** $f(x),f'(x),f''(x)$ continuous on $[0,\ln 2]$, $f(0)=0$, $f'(0)=3$, $f'(\ln 2)=4$, $f(\ln 2)=6$, $\int_0^{\ln 2}e^{-2x}f(x)dx=3$.
$\displaystyle\int_0^{\ln 2}e^{-2x}f''(x)\,dx=$
Step-by-Step Solution
Key Concept: IBP: $\int_0^{\ln 2}e^{-2x}f''dx=[e^{-2x}f']_0^{\ln 2}+2\int_0^{\ln 2}e^{-2x}f'dx$. Apply IBP again: $=[e^{-2x}f']_0^{\ln 2}+2[e^{-2x}f]_0^{\ln 2}+4\int_0^{\ln 2}e^{-2x}fdx$.
$\mathbf{13}$.
Correct Answer: 13