Question

How do you find the most general antiderivative of the function for $f(x) = x - 7$ ?

Answer 1

As we know that an antiderivative of a function $f(x)$ is a function whose derivative is equal to $f(x)$ i.e. if $F'(x) = f(x)$ then $F'(x)$ is an antiderivative of $f(x)$. To find the antiderivative we often reverse the process of differentiation. The general antiderivative of $f(x)$ is $F(x) + c$ , where $F$is a differentiable function, which means that to find antiderivative we have to reverse the
process of finding a derivative.

Complete step by step solution:
Here we are taking the indefinite integral of $f(x)$ which means that $\int {x - 7dx}$ .
We know that properties of integral say that we can break it up in pieces in cases of addition and subtraction , thus,
$\int {x - 7dx = \int {xdx} } - \int {7dx}$ , By further using the property of integrals we get $\int {x - 7dx} = \int {xdx} - 7\int {dx}$, now we first solve $\int {xdx}$.
Using the power rule we multiply the expression by the exponent and then reduce the exponent by one which gives us $2*\dfrac{{{x^{2 - 1}}}}{2} = x$.
So our first integral reduces to $\dfrac{{{x^2}}}{2} + c$ here “c” is a constant number because we are finding the antiderivative.
Now we evaluate $7\int {dx}$ , this is called a perfect integral because its result is $x$. So we have $7$ with it, our final result is $7x + c$, now by putting all them together we get,
$\int x - 7dx = (\dfrac{{{x^2}}}{2} + c) - 97x + c)$ it reduces to$\dfrac{{{x^2}}}{2} + c - 7x - c$, by distributing the negative sign.
Hence the antiderivative of $x - 7 = \dfrac{{{x^2}}}{2} - 7x + c$ .

Note: We should always keep in mind that $c - c \ne 0$, because there are constants and we do not know what another number is there in our antiderivative. Infact $c - c$ will always be a constant and since $c$ represents a constant, we can just call it normal $c$. While calculating antiderivative we should never forget $c$as our final answer always has it.