Definite Integration
Partial Fractions in Definite Integral
nta_pyq_2023_jan
Grade None
Question:
The integral $16\displaystyle\int_1^2\dfrac{dx}{x^3(x^2+2)^2}$ is equal to:
$\dfrac{11}{6}+\log_e 4$
$\dfrac{11}{12}+\log_e 4$
$\dfrac{11}{12}-\log_e 4$
$\dfrac{11}{6}-\log_e 4$
Step-by-Step Solution
Key Concept: Partial fractions: $\frac{1}{x^3(x^2+2)^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x^3}+\frac{Dx+E}{x^2+2}+\frac{Fx+G}{(x^2+2)^2}$. Integrate each term.
$16\int_1^2\frac{dx}{x^3(x^2+2)^2}=\dfrac{11}{6}-\log_e 4$.
Correct Answer: 4