What is the antiderivative of #f(x) = x - 7#?

Question

Emma Aldridge · Accepted Answer

#x^2/2-7x+C# The antiderivative of #f(x)=x-7# is quite simple. It just involves two cases of the reverse power rule. First, we write down the problem in math notation: #intx-7dx# The sum rule for integrals (antiderivatives) says we can simplify this to: #intxdx-int7dx# Furthermore, another rule of integrals says that #intaxdx=aintxdx#; in other words, we can pull constants out of the integral. Since the #7# in the second integral is constant, we can say: #intxdx-7intdx# Solving these is easy. The power rule for derivatives says that to find the derivative of #x^a#, multiply #x# by #a# and then decrease #a# by one (such as #d/dx(x^2)=2*x^(2-1)=2x^1=2x#). It makes sense, then, that the antiderivative would be this in reverse: increase #a# by one, then divide by the new power (so the antiderivative of #2x# would be #2x^(1+1)=2x^2=(2x^2)/2=x^2#, which is the original function). Applying this to our problem: #intxdx=x^2/2+C_1# and #-7intdx=-7x-C_2-># this one is the so-called "perfect integral" Our solution is therefore: #x^2/2+C_1-7x-C_2# And since #C_1-C_2# is just another constant, we can combine them into a general constant #C# to get our final result of: #x^2/2+7x+C#

Scarlett Allison · Answer

undefined The antiderivative of $ f(x) = x - 7 $ is $ F(x) = \frac{x^2}{2} - 7x + C $, where $ C $ is the constant of integration.