45. A curve passing through point \( (1,-3) \) has a gradient of \( \frac{4(2 x+5)^{3}}{3} \). Find the equation of the curve.

To find the equation of the curve, you start by integrating the gradient function. Given the gradient \( \frac{4(2x+5)^{3}}{3} \), we integrate with respect to \( x \): \[ y = \int \frac{4(2x + 5)^{3}}{3} \, dx \] Using substitution: let \( u = 2x + 5 \), then \( du = 2dx \) or \( dx = \frac{du}{2} \). The limits change when substituting as well: \[ y = \int \frac{4u^{3}}{3} \cdot \frac{du}{2} = \frac{2}{3} \int u^{3} \, du = \frac{2}{3} \cdot \frac{u^{4}}{4} + C = \frac{u^{4}}{6} + C \] Substituting back for \( u \): \[ y = \frac{(2x + 5)^{4}}{6} + C \] Now, we need to find \( C \) using the point \( (1, -3) \): \[ -3 = \frac{(2(1) + 5)^{4}}{6} + C \implies -3 = \frac{7^{4}}{6} + C \] Calculating \( 7^{4} = 2401 \): \[ -3 = \frac{2401}{6} + C \implies C = -3 - \frac{2401}{6} = -\frac{18}{6} - \frac{2401}{6} = -\frac{2419}{6} \] Thus, the equation of the curve is: \[ y = \frac{(2x+5)^{4}}{6} - \frac{2419}{6} \]