Question

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

∫ (x + 8x) dx

Answer 1

Step-by-step explanation:

Given

$\int\limits {(x+8x)} \, dx$  

to integrate we use

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

we have

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

$y= \frac{x^1^+^1}{1+1}+ \frac{8x^1^+^1}{1+1}+c\\\\y= \frac{x^2}{2}+ \frac{8x^2}{2}+c\\\\y= \frac{x^2}{2}+ 4x^2+c\\\\y= \frac{x^2+8x^2}{2}+c\\\\y= \frac{9x^2}{2}+c$

in order to verify using differentiation we use

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

$y= \frac{9x^2}{2}+c\\\\ \frac{dy}{dx} =2* \frac{9x^2^-^1}{2}\\\\ \frac{dy}{dx} =2* \frac{9x}{2}\\\\ \frac{dy}{dx} =9x\\\\9x= x+8x\\\\ \frac{dy}{dx}= x+8x$