# Find the antiderivative, F(x), with F'(x) = f(x) and F(0) = 1. f(x) = -7 x.

## Question:

Find the antiderivative, F(x), with F'(x) = f(x) and F(0) = 1.

f(x) = -7 x.

## Integration:

Integration is defined as the total area under given curve.

Integration is also called antiderivative.

In this question we need to differentiate f(x) in order to obtain F(x) with initial value condition.

After getting f(x) use initial value condition to get c and hence obtain actual value of F(x).

We have,

f(x) = -7 x

F'(x) = f(x)

F(0) = 1

now,

{eq}F(x) = \int F'(x) \\ F(x) = \int f(x) \\ {/eq}

also,

f(x) = -7x

{eq}F(x) = \int -7x \\ F(x) = -7 (\frac{x^2}{2}) + c {/eq}

now,

for F(0) = 1

{eq}F(0) = -7 (\frac{(0)^2}{2}) + c \\ \implies c = 1 {/eq}

so,

{eq}\therefore \color{blue}{F(x) = \frac{-7 \ x^2}{2} + 1 } {/eq}