Exercise 7.4 - Question 16
Problem
$$\int \frac{4x+1}{\sqrt{2x^2 + x - 3}} \, dx$$
Step-by-Step Solution
Step 1: Express Numerator in Terms of Derivative
d/dx(2x² + x - 3) = 4x + 1
Notice: The numerator IS exactly the derivative of the expression under the root!
Step 2: Use Substitution
Let u = 2x² + x - 3, then du = (4x + 1)dx
$$\int \frac{4x+1}{\sqrt{2x^2 + x - 3}} \, dx = \int \frac{du}{\sqrt{u}}$$
Step 3: Integrate
$$\int u^{-1/2} \, du = 2u^{1/2} + C = 2\sqrt{u} + C$$
Step 4: Substitute Back
$$= 2\sqrt{2x^2 + x - 3} + C$$
✅ Final Answer
$$\int \frac{4x+1}{\sqrt{2x^2 + x - 3}} \, dx = 2\sqrt{2x^2 + x - 3} + C$$