Find the function $F$ such that

$\frac{dF}{dx}=-7x+31$

and $F(5)=22$

Our goal: find $F.We$ have $to$ integrate and then substitute$-$and$-$solve $to$ find $it.$

Write down the differential equation

$On$ your paper, write down the given differential equation, $\frac{dF}{dx}=-7x+31.$

Integrate

Multiply both sides $bydx.$ You should have $dF=(-7x+31)dx$

Integate both sides. Get $F={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(-7x+31)dx$

Now actually $do$ the integration $off$ and find the most general antiderivative.

$F_{\text{g}\text{e}\text{n}}(x)=$

Find the value $of$ the constant $of$ integration

$We$ were given $F(5)=22.$ This $is$ a point $on$ the curve.

$(6)$ Plug $x=5$ into the general $F_{\text{g}\text{e}\text{n}}(x),$ set $it$ equal $toF(5)=22,$ and solve $to$ find $C.$

$C=$

Conclude

We've found the general solution and the constant $of$ integration. $We$ just need $to$ package them together, and find the specific function $F$ going through the given point.

Use the value $ofC$ with your general $F_{\text{g}\text{e}\text{n}},to$ form the final $F.$

$F(x)=$