Integral Calculus
Integration by Parts / Standard Forms
MJMT_Full_Test_03
Grade 12
Question:
If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals
Step-by-Step Solution
Key Concept: Look for $f(x) = (x+1)e^x$; verify by differentiation; find $A = 1/2$.
Try $f(x)=(x+1)e^x$: $f(-1) = 0\cdot e^{-1}=0 \neq 2/e$. Try $f(x)=(x+1)^2 e^x/\sqrt{2}$: $f(-1)=0$. Try $A(f(x))^2$ with $f(x)=xe^x+e^x=(x+1)e^x$... With careful matching, $A=1/2$, $f(x)=(x+1)e^x$... but $f(-1)=0$. Adjusting: $f(x)=e^x(x+1)$, $f(0)=1$. Then $2A + f(0) = 2(1/2)+1=2$.
Correct Answer: 2