Evaluate ∫ 2, 1 (( 2x^2 + √x−1) / x^2)dx.

Question

Evaluate ∫ ²₁ (( 2x² + √x−1) / x²)dx.

Answer 1

When I taught at the Naval Academy, I always encouraged students to "do the math first" before integration. That idea holds true here. If you look at the equation (2x^2 + x^(1/2) - 1)/*x^2), you can break it into three parts (aka, the math first) 2 + x^(-3/2) - x^(-2). Now each can be integrated via the power rule or

2x -2x^(-1/2) + x^(-1). Then evaluate at 2, evaluate at 1 and take the difference

(4-sqrt(2) + 1/2) - (2-2+1)

(4-sqrt(2) + 1/2) - 1

(4-sqrt(2) -1/2

3.5 - sqrt(2)