Question

Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)

Answer 1

Answer:

$y = \frac{9x^2}{2} + c$

Step-by-step explanation:

Given

$\int\limits^ _$$(x + 8x) dx$

Required

(a) Integrate

(b) Check using differentiation

To integrate, we make use of the following formula;

if

$\frac{dy}{dx} = \int\limits^{} _{} ax^n$

then

$y = \frac{ax^{n+1}}{n+1}$

So; $\int\limits^ _$$(x + 8x) dx$ becomes

$y = \frac{x^{1+1}}{1+1} + \frac{8x^{1+1}}{1+1} + c$

$y = \frac{x^{2}}{2} + \frac{8x^{2}}{2} + c$

$y = \frac{x^{2}}{2} + 4x^2 + c$

Take LCM

$y = \frac{x^{2} + 8x^2}{2} + c$

$y = \frac{9x^2}{2} + c$

To check using differentiation, we make use of

if $y = ax^n$, then

$\frac{dy}{dx} = nax^{n-1}$

Using this formula

$y = \frac{9x^2}{2} + c$ becomes

$\frac{dy}{dx} = 2 * \frac{9x^{2-1}}{2}$

$\frac{dy}{dx} = 2 * \frac{9x}{2}$

$\frac{dy}{dx} =9x$

$9x = x + 8x$

So;

$\frac{dy}{dx} = x + 8x$