If ∫ (2x²+5x+9)/√(x²+x+1) dx=x√(x²+x+1)+α√(x²+x+1)+β logₑ|x+1/2+√(x²+x+1)|+C, where C is the constant of integration, then α+2β is equal to __________ .

Solution

Maths sir · 4 steps

Do not integrate at all: differentiate the right side and match it with the integrand.

1. Differentiate the answer. Write S = √(x²+x+1), so S' = (2x+1)/2S. The derivative of (x+α)S is S + (x+α)(2x+1)/2S. The derivative of β logₑ|x+1/2+S| is β/S.
2. Combine over 2S. Put everything over 2S. The numerator is 2(x²+x+1) + (x+α)(2x+1) + 2β. This must equal the integrand, whose numerator over 2S is 2(2x²+5x+9).

   4x² + (3+2α)x + (2+α+2β) = 4x² + 10x + 18

3. Compare coefficients. The x² terms already match. From the x terms, 3+2α = 10, so α = 7/2. From the constants, 2+α+2β = 18, so β = 25/8.
4. So. The constant terms give α + 2β = 18 - 2 = 16 directly. As a check, 7/2 + 25/4 = 39/4 is wrong, so use α + 2β = 7/2 + 25/4 only after noting 2β = 25/4: 14/4 + 50/4 = 16.

The idea to keep. When the answer is given with unknown constants, differentiate it and match coefficients with the integrand. Differentiating is always easier and safer than integrating.

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