Integral Calculus
Partial Fractions
MMTS_Full_Test_08
Grade 12
Question:
$\displaystyle\int\dfrac{2x^3+3x^2-11x-17}{(x^2-x-6)^2}dx$
$\dfrac{A}{x-3}+\dfrac{B}{x+2}+C\ln|x-3|+D\ln|x+2|+E$
$\dfrac{A}{(x-3)^2}+\dfrac{B}{x-3}+\dfrac{C}{(x+2)^2}+\dfrac{D}{x+2}+E$
$\dfrac{A}{x^2-x-6}+\dfrac{B}{x-3}+\dfrac{C}{x+2}+E$
$\dfrac{Ax+B}{x^2-x-6}+C\ln|x-3|+D\ln|x+2|+E$
Step-by-Step Solution
Key Concept: Denominator $(x-3)^2(x+2)^2$; partial fractions have terms up to degree 2 for each repeated factor
Partial fractions: $\frac{A}{x-3}+\frac{B}{(x-3)^2}+\frac{C}{x+2}+\frac{D}{(x+2)^2}$. Option 2.
Correct Answer: 2