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Express the numerator 2x2 + 5x + 4 in the form A(x2+x+1) + B(2x+1) + C.

Mathematics

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Answer

To evaluate the integral ∫2x2+5x+4x2+x+1 dx\int \frac{2x^2 + 5x + 4}{\sqrt{x^2 + x + 1}} \, dx, we use the method of expressing the numerator in terms of the denominator and its derivative.

Step 1: Express the numerator 2x2+5x+42x^2 + 5x + 4 in the form A(x2+x+1)+B(2x+1)+CA(x^2+x+1) + B(2x+1) + C.
Let 2x2+5x+4=A(x2+x+1)+B(2x+1)+C2x^2 + 5x + 4 = A(x^2+x+1) + B(2x+1) + C.
Expanding the right side:
$2x^2 + 5x + 4 = Ax^2 + Ax + A +

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