Question

If \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] then \( \frac{2}{3} (A + B + C + D) = \)}

• A. \( \frac{2}{3} \)
• B. \( -\frac{2}{3} \)
• C. \( \frac{4}{3} \)
• D. \( -\frac{4}{3} \)

Hint:

When solving integrals involving square roots, use substitution and standard integration formulas to break the problem into manageable parts.

Answer 1

The Correct Option is B

To solve the integral and find the value of \(\frac{2}{3}(A + B + C + D)\), we follow these steps: Given Integral: \[ \int \frac{\sqrt[4]{x}}{\sqrt{x} + \sqrt[4]{x}} \, dx = \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] Step 1: Simplify the Integral Let’s make a substitution to simplify the integral. Let \( u = \sqrt[4]{x} \). Then, \( x = u^4 \) and \( dx = 4u^3 \, du \). Substituting into the integral: \[ \int \frac{u}{\sqrt{u^4} + u} \cdot 4u^3 \, du = \int \frac{u}{u^2 + u} \cdot 4u^3 \, du = \int \frac{4u^4}{u^2 + u} \, du \] Simplify the integrand: \[ \frac{4u^4}{u^2 + u} = \frac{4u^4}{u(u + 1)} = \frac{4u^3}{u + 1} \] Step 2: Perform Polynomial Division Divide \( 4u^3 \) by \( u + 1 \): \[ 4u^3 = 4u^2(u + 1) - 4u^2 \] \[ -4u^2 = -4u(u + 1) + 4u \] \[ 4u = 4(u + 1) - 4 \] So, \[ \frac{4u^3}{u + 1} = 4u^2 - 4u + 4 - \frac{4}{u + 1} \] Step 3: Integrate Term by Term Integrate each term separately: \[ \int \left( 4u^2 - 4u + 4 - \frac{4}{u + 1} \right) du = \frac{4}{3}u^3 - 2u^2 + 4u - 4 \ln|u + 1| + K \] Step 4: Substitute Back \( u = \sqrt[4]{x} \) \[ \frac{4}{3} (\sqrt[4]{x})^3 - 2 (\sqrt[4]{x})^2 + 4 \sqrt[4]{x} - 4 \ln|1 + \sqrt[4]{x}| + K \] Step 5: Compare with Given Form The given form is: \[ \frac{2}{3} \left[ A \sqrt[4]{x^3} + B \sqrt[4]{x^2} + C \sqrt[4]{x} + D \log \left( 1 + \sqrt[4]{x} \right) \right] + K \] Comparing coefficients: \[ \frac{2}{3} A = \frac{4}{3} \Rightarrow A = 2 \] \[ \frac{2}{3} B = -2 \Rightarrow B = -3 \] \[ \frac{2}{3} C = 4 \Rightarrow C = 6 \] \[ \frac{2}{3} D = -4 \Rightarrow D = -6 \] Step 6: Calculate \( \frac{2}{3}(A + B + C + D) \) \[ A + B + C + D = 2 - 3 + 6 - 6 = -1 \] \[ \frac{2}{3}(A + B + C + D) = \frac{2}{3} \times (-1) = -\frac{2}{3} \] Final Answer: \[ \boxed{-\frac{2}{3}} \]