Question

If f(x) = ∫$_{x}^{1}$ (5x⁸ + 7x⁶)/(x² + 1 + 2x⁷)² dx, (x ≥ 0), f(0) = 0 and f(1) = 1/K, then the value of K is __________.

Hint:

In integrals with high powers in the denominator, try dividing the numerator and denominator by $x^n$ to find a substitution.

Answer 1

Correct Answer: 4

Step 1: Divide numerator and denominator by $x^{14}$: $f(x) = \int \frac{5x^{-6} + 7x^{-8}}{(x^{-5} + x^{-7} + 2)^2} dx$.
Step 2: Let $u = x^{-5} + x^{-7} + 2$. Then $du = (-5x^{-6} - 7x^{-8}) dx$.
Step 3: The integral becomes $\int \frac{-du}{u^2} = \frac{1}{u} + C = \frac{1}{x^{-5} + x^{-7} + 2} + C$.
Step 4: $f(x) = \frac{x^7}{1 + 2x^7 + x^2} + C$. (Using boundary conditions provided).
Step 5: For $f(1) = \frac{1}{1+2+1} = \frac{1}{4}$.
Step 6: $1/K = 1/4 \implies K = 4$.