Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
If $\int \frac{dx}{x^2\left(x^7 - 6\right)} = A\left[\ln\left(p^9 + 9p^2 - 2p^3 - 18p\right)\right] + c$, then:
$A = \frac{1}{9072}$
$p = \left(\frac{x^7-6}{x^7}\right)$
$A = \frac{1}{54432}$
$p = \left(\frac{x^7-6}{x^7}\right)^{-1}$
Step-by-Step Solution
Key Concept: Strategic substitution transforms a complex rational function into a polynomial integrand that can be expanded and integrated term by term.
We use the substitution $1 - \frac{6}{x^7} = p$, which gives $\frac{42}{x^8}dx = dp$ and $x^7 = \frac{6}{1-p}$. The integral becomes $I = \frac{1}{42}\int \frac{(1-p)^7}{(6)^7}dp$, which after expansion and integration yields $I = \frac{1}{54432}[\ln p^6 + 9p^2 - 2p^3 - 18p] + c$.
Correct Answer: 3,2