The general solution is obtained by integrating and adding a constant \(C\):

Example: Given \( \dfrac{dy}{dx} = 6x^2 \), find \(y\).

\[ y = \int 6x^2 dx = 2x^3 + C \] This is the general solution.

The particular solution is found by using a given point to calculate \(C\).

Example: Given \( \dfrac{dy}{dx} = 6x^2 \) and \(y=8\) when \(x=1\), find \(y\).

First integrate: \[ y = 2x^3 + C. \] Substitute \(x=1, y=8\): \[ 8 = 2(1)^3 + C \Rightarrow C = 6. \] Final answer: \[ y = 2x^3 + 6. \]

Before integrating, rewrite as powers:

• \(\sqrt{x} = x^{1/2}\)
• \(\sqrt[3]{x} = x^{1/3}\)
• \(\frac{1}{x^3} = x^{-3}\)

Example 1: General solution \(\displaystyle \frac{dy}{dx} = \sqrt{x}\)

Rewrite: \(x^{1/2}\) \[ y = \int x^{1/2} dx = \frac{2}{3}x^{3/2} + C \]

Example 2: Particular solution \(\displaystyle \frac{dy}{dx} = \frac{4}{x^2}, \ y = 7 \text{ when } x = 1\)

Rewrite: \( \frac{4}{x^2} = 4x^{-2} \) \[ y = \int 4x^{-2} dx = -4x^{-1} + C = -\frac{4}{x} + C \] Substitute \(x=1, y=7\): \[ 7 = -4 + C \Rightarrow C = 11. \] Final equation: \[ y = -\frac{4}{x} + 11. \]

Given: \( \dfrac{dy}{dx} = 3x^2 - 4x^{1/2} \), and the curve passes through \((1, 5)\).

Step 1: Integrate. \[ y = \int (3x^2 - 4x^{1/2})dx = x^3 - \frac{8}{3}x^{3/2} + C \] Step 2: Use the point \((1, 5)\): \[ 5 = 1 - \frac{8}{3}(1) + C \Rightarrow C = \frac{20}{3} \] Step 3: Final equation: \[ y = x^3 - \frac{8}{3}x^{3/2} + \frac{20}{3} \]

• Always rewrite roots and fractions as powers before integrating.
• Always include +C for general solutions.
• Never include +C for definite integrals or area/volume problems.
• To find C, always substitute into the integrated function.
• Do NOT differentiate accidentally — integrate carefully.