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

Solution

Maths sir · 4 steps

Pull x⁷ out of the bracket so the numerator becomes the derivative of what is left inside.

1. Divide by x^14. Divide the numerator and the denominator by x¹⁴. The bracket squared becomes (2 + x⁻⁵ + x⁻⁷)² and the numerator becomes 5x⁻⁶ + 7x⁻⁸.
2. Substitute. Let t = 2 + x⁻⁵ + x⁻⁷. Then dt = (-5x⁻⁶ - 7x⁻⁸)dx, which is exactly minus the numerator.

   f(x) = -∫ dt/t² = 1/t + C = x⁷/(2x⁷ + x² + 1) + C

3. Fix the constant. At x = 0 the fraction is 0, so f(0) = 0 gives C = 0.
4. So. f(1) = 1/(2 + 1 + 1) = 1/4, so K = 4.

The common slip: -4. Dropping the minus sign in dt = -(5x⁻⁶ + 7x⁻⁸)dx gives f(1) = -1/4, so K = -4.

The idea to keep. When the denominator is a squared polynomial, divide everything by the power of x that makes the numerator the derivative of the bracket. Then ∫ dt/t² finishes it.

Answer from the NTA final answer key. Solution by PrepWiser.