If we know the derivative of a function, we can find the equation of the curve by integration. We find either a general solution or a particular solution.
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:
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: \[ y = x^3 - \frac{8}{3}x^{3/2} + \frac{20}{3} \]