Question 442217
*[tex \LARGE \int 4(2x+5)^3\,dx]


Begin with the substitution u = 2x + 5. Since *[tex \frac{du}{dx} = 2], then this implies that *[tex du = 2dx] and *[tex dx = \frac{du}{2}]. We substitute to obtain an integral in terms of u:


*[tex \LARGE \int 4u^3\, (\frac{du}{2}) = \frac{1}{2} \int 4u^3\, du]


This is easily integrable, using the power rule:


*[tex \LARGE \frac{1}{2} \int 4u^3\, du = \frac{1}{2} 4(\frac{u^4}{4}) + C = \frac{u^4}{2} + C] where C is a constant. Since we want our expression in terms of x instead of in terms of u, we back substitute to get


*[tex \LARGE \int 4(2x+5)^3\, dx = \frac{(2x+5)^4}{2} + C]