<p>Evaluate: \ 5\int \frac{dx}{\sqrt{1+x} - \sqrt[3]{1+x}}</p>
Step-by-Step Solution
Key Concept: Use substitution u = (1+x)^(1/6) to convert roots to integer powers, transforming the irrational integrand into a rational function that can be integrated using partial fractions or polynomial division.
<p><strong>Step 1:</strong> Let u = (1+x)^(1/6), so 1+x = u^6 and dx = 6u^5 du</p><p><strong>Step 2:</strong> Rewrite the integrand in terms of u:<br/>√(1+x) = u^3 and ∛(1+x) = u^2<br/>The integral becomes: ∫[6u^5/(u^3 - u^2)] du = 6∫[u^5/(u^2(u-1))] du = 6∫[u^3/(u-1)] du</p><p><strong>Step 3:</strong> Perform polynomial division of u^3 by (u-1):<br/>u^3/(u-1) = u^2 + u + 1 + 1/(u-1)</p><p><strong>Step 4:</strong> Integrate term by term:<br/>6∫[u^2 + u + 1 + 1/(u-1)] du = 6[u^3/3 + u^2/2 + u + ln|u-1|] + C<br/>= 2u^3 + 3u^2 + 6u + 6ln|u-1| + C</p><p><strong>Step 5:</strong> Substitute back u = (1+x)^(1/6):<br/>= 2(1+x)^(1/2) + 3(1+x)^(1/3) + 6(1+x)^(1/6) + 6ln|(1+x)^(1/6) - 1| + C</p><p><strong>Step 6:</strong> Multiply by 5 (from the original integral):<br/>∴ <strong>Answer: 10√(1+x) + 15∛(1+x) + 30(1+x)^(1/6) + 30ln|(1+x)^(1/6) - 1| + C</strong></p>
Correct Answer: 10