Integral Calculus
Integration with substitution; identifying α+β+γ
MMTS_Full_Test_20
Grade 12
Question:
Let $I(x)=\displaystyle\int\frac{x+1}{x(x^2e^{2x}-1)}\,dx=\frac{1}{4}\ln\frac{(xe^x)^\alpha-2(xe^x)^\beta+\gamma}{x^4e^{4x}}+c$, then $\alpha+\beta+\gamma$ equals
Step-by-Step Solution
Key Concept: Let $u=xe^x$. Then $(x+1)dx=du/e^x\cdot e^x/(1)=... $ Use partial fractions on $1/(u^2-1)=1/((u-1)(u+1))$.
$\alpha+\beta+\gamma=4+2+1=7$.
Correct Answer: 7