If \(\displaystyle \int \frac{x\,dx}{\sqrt[15]{(1+x^2)^{12}(2+x^2)^{18}}}=\alpha\left(\frac{1+x^2}{2+x^2}\right)^{1/n}+C\), then \(\dfrac{n}{\alpha}=\)
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When the integrand contains powers of \(1+x^2\) and \(2+x^2\), try the substitution
\[
u=\frac{1+x^2}{2+x^2}.
\]
This substitution is useful because its derivative contains
\[
\frac{x\,dx}{(2+x^2)^2}.
\]
Step 1: Rewrite the radical in exponent form. The 15th root becomes exponent 1/15: (1+x²)^(12/15) (2+x²)^(18/15) = (1+x²)^(4/5) (2+x²)^(6/5). Integral = ∫ x dx / [(1+x²)^(4/5) (2+x²)^(6/5)]. Step 2: Factor and substitute cleverly. Write as ∫ x dx / [(2+x²)² ((1+x²)/(2+x²))^(4/5)]. Let t = (1+x²)/(2+x²). Then dt = [2x(2+x²) – 2x(1+x²)]/(2+x²)² dx = 2x/(2+x²)² dx. So x dx/(2+x²)² = dt/2. Step 3: Simplify and integrate. Integral becomes ∫ (1/t^(4/5)) · (dt/2) = (1/2) ∫ t^(–4/5) dt = (1/2)·5·t^(1/5) + C = (5/2) t^(1/5) + C. Step 4: Back-substitute and compare. Result = (5/2)[(1+x²)/(2+x²)]^(1/5) + C. Comparing with α[(1+x²)/(2+x²)]^(1/n) gives α = 5/2, n = 5. Step 5: Final Answer: n/α = 5/(5/2) = 2.