Integrate, add a constant, then substitute the given point to determine it.
Example: \(y\prime=3x^2+2\), through \((1,6)\)
| Step | Working |
|---|---|
| Integrate and add \(c\) | |
| Substitute \((1,\,6)\) | |
| Solve for \(c\) | |
| State the curve |
Using a known derivative
\(g\prime(x)=h(x)\quad\Longrightarrow\quad\int kh(x)\,\mathrm dx=kg(x)+C\)
For a definite integral, \(\int_a^bh(x)\,\mathrm dx=g(b)-g(a)\), where the integral exists.
Second derivatives
| Given | Integrate | Fixing the constants |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and a point on the curve | ||
| \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\), the gradient at a point and a point on the curve |
Each integration introduces a new constant. Gradient information fixes the first; a point fixes the second. Disconnected domain intervals may have independent constants.
Need help? Join our JC Math tuition classes.
Learn more