Indefinite Integration
Integral giving f(x) — properties of solution curves
MJAT_TS5_P2
Grade 12
Question:
If $\displaystyle\int\frac{x+2}{x^4+8x^3+20x^2+16x+4}\,dx = f(x)+C$ and $f(0)=\dfrac{\ln 2}{2}$, then which is/are INCORRECT?
A) Number of solutions of $f(x)=x^2$ is $2$
B) Number of solutions of $f(x)=x^2$ is $1$
C) Number of solutions of $f(x)=4-x^2$ is $2$
D) Number of solutions of $f(x)=4-x^2$ is $1$
Step-by-Step Solution
Key Concept: Factor denominator: $x^4+8x^3+20x^2+16x+4=(x^2+4x+2)^2$. So integrand $=\frac{x+2}{(x^2+4x+2)^2}=\frac{1}{2}\cdot\frac{2x+4}{(x^2+4x+2)^2}$. Integral $=\frac{-1}{2(x^2+4x+2)}+C$. With $f(0)=\ln 2/2$: $-1/4+C=\ln 2/2$... Hmm, that doesn't match. Perhaps $f(x)=\frac{1}{2}\ln|x^2+4x+2|+C$.
B ✗, C ✗. INCORRECT: B, C.
Correct Answer: BC