find the indefinite integral. (remember the constant of integration. remember to use absolute values where…
find the indefinite integral. (remember the constant of integration. remember to use absolute values where appropriate.) ∫(x^8 / (x^9 - 2)) dx
Answer
Explanation:
Step1: Use substitution
Let $u = x^{9}-2$, then $du=9x^{8}dx$, and $x^{8}dx=\frac{1}{9}du$.
Step2: Rewrite the integral
The integral $\int\frac{x^{8}}{x^{9}-2}dx$ becomes $\frac{1}{9}\int\frac{du}{u}$.
Step3: Integrate
We know that $\int\frac{1}{u}du=\ln|u| + C$. So, $\frac{1}{9}\int\frac{du}{u}=\frac{1}{9}\ln|u|+C$.
Step4: Substitute back
Substitute $u = x^{9}-2$ back into the result, we get $\frac{1}{9}\ln|x^{9}-2|+C$.
Answer:
$\frac{1}{9}\ln|x^{9}-2|+C$