Answer:
[tex]y = \frac{9x^2}{2} + c[/tex]
Step-by-step explanation:
Given
[tex]\int\limits^ _[/tex][tex](x + 8x) dx[/tex]
Required
(a) Integrate
(b) Check using differentiation
To integrate, we make use of the following formula;
if
[tex]\frac{dy}{dx} = \int\limits^{} _{} ax^n[/tex]
then
[tex]y = \frac{ax^{n+1}}{n+1}[/tex]
So; [tex]\int\limits^ _[/tex][tex](x + 8x) dx[/tex] becomes
[tex]y = \frac{x^{1+1}}{1+1} + \frac{8x^{1+1}}{1+1} + c[/tex]
[tex]y = \frac{x^{2}}{2} + \frac{8x^{2}}{2} + c[/tex]
[tex]y = \frac{x^{2}}{2} + 4x^2 + c[/tex]
Take LCM
[tex]y = \frac{x^{2} + 8x^2}{2} + c[/tex]
[tex]y = \frac{9x^2}{2} + c[/tex]
To check using differentiation, we make use of
if [tex]y = ax^n[/tex], then
[tex]\frac{dy}{dx} = nax^{n-1}[/tex]
Using this formula
[tex]y = \frac{9x^2}{2} + c[/tex] becomes
[tex]\frac{dy}{dx} = 2 * \frac{9x^{2-1}}{2}[/tex]
[tex]\frac{dy}{dx} = 2 * \frac{9x}{2}[/tex]
[tex]\frac{dy}{dx} =9x[/tex]
[tex]9x = x + 8x[/tex]
So;
[tex]\frac{dy}{dx} = x + 8x[/tex]
