Question

$\int\frac{dx}{x^{\frac{1}{2}}+x^{\frac{1}{3}}}=Ax^{\frac{1}{2}}+Bx^{\frac{1}{3}}+Cx^{\frac{1}{6}}+D~log(x^{\frac{1}{6}}+1)+k$, then values of A, B, C and D are}

• A. 2, -3, 6, -6
• B. 2, 3, -6, 6
• C. 2, -3, -6, 6
• D. -2, -3, 6, 6

Hint:

For integrals with $x^{1/n}$ and $x^{1/m}$, substitute $x = t^{LCM(n, m)}$.

Answer 1

The Correct Option is A

Step 1: Concept
Use substitution $x = t^{6}$ to eliminate fractional powers.

Step 2: Analysis
$dx = 6t^{5}dt$. The integral becomes $\int\frac{6t^{5}}{t^{3}+t^{2}}dt = \int\frac{6t^{3}}{t+1}dt$.
Using long division: $6\int(t^{2}-t+1-\frac{1}{t+1})dt$.

Step 3: Calculation
$6(\frac{t^{3}}{3} - \frac{t^{2}}{2} + t - log(t+1)) = 2t^{3} - 3t^{2} + 6t - 6~log(t+1)$.
Substitute back $t = x^{1/6}$: $2x^{1/2} - 3x^{1/3} + 6x^{1/6} - 6~log(x^{1/6}+1)$.

Step 4: Conclusion
Comparing coefficients: $A=2, B=-3, C=6, D=-6$. Final Answer: (A)