If $\int \frac{dx}{\sqrt{x + 7} - \sqrt[3]{x + 7}} = P\sqrt{x + 7} + Q\sqrt[3]{x + 7} + R\ln|x + 7|^{1/4}| + c$. Then find the value of $P + Q + R$.
Step-by-Step Solution
Key Concept: Use the substitution $x + 7 = t^4$ to convert radical expressions into polynomial form, then perform polynomial division to simplify the rational integrand.
Given $x + 7 = t^4$, we have $dx = 4t^3dt$. The integral becomes $\int\frac{4t^3dt}{t^2 - t} = 4\int\frac{t^2 + 1 + 1}{t - 1}dt = 4\int\left(t + 1 + \frac{1}{t-1}\right)dt = 4\left(\frac{t^2}{2} + t + \ln|t-1|\right) + c = 2\sqrt[4]{x+7} + 4(x+7)^{1/4} + 4\ln|\sqrt[4]{x+7} - 1| + c$.
Correct Answer: 1,2,3