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ss7ja [257]
3 years ago
15

Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative. f"(x) = 2x + 9e

x Rx) =
Mathematics
1 answer:
ad-work [718]3 years ago
8 0

Answer:

f(x)=\frac{x^3}{3}+9e^x+Cx+D

Step-by-step explanation:

The function <em>F(x)</em> is an antiderivative of the function <em>f(x)</em> on an interval <em>I</em> if

<em>F′(x)</em> = <em>f(x)</em> for all <em>x </em>in <em>I</em>.

The function <em>F(x) + C</em> is the General Antiderivative of the function <em>f(x)</em> on an interval <em>I</em> if <em>F′(x) = f(x)</em> for all <em>x</em> in <em>I </em>and <em>C</em> is an arbitrary constant.

The Indefinite Integral of <em>f(x)</em> is the General Antiderivative of <em>f(x)</em>.

\int {f(x)} \, dx =F(x)+C

To find the first antiderivative you must integrate the function f''(x) = 2x + 9e^x

f'(x)=\int { 2x + 9e^x} \, dx \\\\\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\\int \:2xdx+\int \:9e^xdx = x^2+9e^x\\\\\mathrm{Add\:a\:constant\:to\:the\:solution}\\\\x^2+9e^x+C

To find the second antiderivative you must integrate the function f'(x) =x^2+9e^x+C

f(x)=\int {x^2+9e^x+C} \, dx\\\\\mathrm{Apply\:the\:Sum\:Rule}:\quad \int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm \int g\left(x\right)dx\\\\\int \:x^2dx+\int \:9e^xdx+\int \:Cdx= \frac{x^3}{3}+9e^x+Cx\\\\\mathrm{Add\:a\:constant\:to\:the\:solution}\\\\\frac{x^3}{3}+9e^x+Cx+D

Therefore,

f(x)=\frac{x^3}{3}+9e^x+Cx+D

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