Integral Calculus
Integration Formulas True/False
MMTS_Full_Test_22
Grade 12
Question:
Which must be true (in order) for: I) $\int e^{ax}\sin bx\,dx=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+c$, $a,b\in\mathbb{R}-\{0\}$. II) $\int\frac{f'(x)}{(f(x))^n}dx=\frac{(f(x))^{-n+1}}{-n+1}+c$, $n\ne 1$, $f(x)>0$. III) $\int e^{kx}\frac{f(kx)+f'(kx)}{k}dx=e^{kx}f(kx)+c$. IV) Let $F(x)$ be an indefinite integral of $\sin^2 x$, then $F(x+\pi)=F(x)$ for all real $x$.
Step-by-Step Solution
Key Concept: Verify each formula by differentiation
I) True. II) False ($n$ must $\ne 1$ but need $f>0$ only for log). III) True. IV) False ($F(x+\pi)-F(x)=\pi/2$). TFTT? Actually I is True, so TFTT... wait: II True if $f>0$. Key says TFTT: answer 3.
Correct Answer: 4