AP DSC School Assistant Math Model Test 3

Concept:The given integral can be simplified by factoring the denominator, and then using the standard logarithmic integration formula: $\int \frac{f'(x)}{f(x)}dx = \ln|f(x)|+C$.Explanation:We start with $\int \frac{\pi}{x^{n+1}-x}\,dx$. Factor the denominator: $x^{n+1}-x = x(x^n-1)$. So the integral becomes $$I = \pi \int \frac{1}{x(x^n-1)}\,dx.$$ Now consider the function $f(x)= \frac{x^n-1}{x^n} = 1 - x^{-n}$. Differentiate $f(x)$: $f'(x) = 0 - (-n)x^{-n-1} = \frac{n}{x^{n+1}}$. Notice that $\frac{f'(x)}{f(x)} = \frac{n}{x^{n+1}} \cdot \frac{x^n}{x^n-1} = \frac{n}{x(x^n-1)}$. Thus, $\frac{1}{x(x^n-1)} = \frac{1}{n} \cdot \frac{f'(x)}{f(x)}$. Integrating both sides: $\int \frac{1}{x(x^n-1)}\,dx = \frac{1}{n} \ln|f(x)| + C' = \frac{1}{n} \ln\left|\frac{x^n-1}{x^n}\right| + C'$. Multiplying by $\pi$ gives $$I = \frac{\pi}{n} \ln\left|\frac{x^n-1}{x^n}\right| + C.$$ This matches option A.Answer:Option A: $\frac{\pi}{n} \log_{e}\left|\frac{x^{n}-1}{x^{n}}\right|+C$