Section 4.8 Antiderivatives
Definition 4.8.1. Antiderivative.
A function \(F\) is an antiderivative of \(f\) on an interval \(I\) if \(F'(x)=f(x)\) for all \(x\) in \(I\text{.}\)
Example 4.8.2.
Find antiderivatives for:
\(\displaystyle f(x)=3x^2\)
\(\displaystyle g(x)=\cos x\)
\(h(x)= \frac{1}{x}\) (contrast the natural domain and the domain \((0,\infty)\))
\(\displaystyle x^3\)
\(\displaystyle \sin x\)
\(\displaystyle \ln |x|\)
Theorem 4.8.3.
If \(F\) is any antiderivative of \(f\) on an interval \(I\text{,}\) then the most general antiderivative is \(F(x)+C\) where \(C\) is an arbitrary constant.
Example 4.8.4.
Find the general antiderivative of \(f(x)=xe^{x^2}\text{.}\)
\(\frac12 e^{x^2} + C\)
Example 4.8.5.
Find an antiderivative \(F\) of \(f(x)=\sin(4x)\) that satisfies \(F(0)=-1\text{.}\)
\(F(x) = -\frac14 \cos (4x) -\frac34\)
Subsection 4.8.1 Indefinite integrals
Definition 4.8.6. Indefinite integral.
The collection of all antiderivatives of a function \(f(x)\) is called the indefinite integral of \(f\) with respect to \(x\) and is denoted
The function \(f(x)\) is called the integrand and the differential \(dx\) indicates that \(x\) is the variable of integration.
Fact 4.8.7. Linearity Rules.
The indefinite integral obeys the same rules with regard to scalar multiplication and addition that the derivative does:
\(\displaystyle \int kf(x)\, dx = k \int f(x)\, dx\)
\(\displaystyle \int -f(x)\, dx = -\int f(x)\, dx\)
\(\displaystyle \int f(x) \pm g(x)\, dx = \int f(x)\, dx \pm \int g(x)\, dx\)
Example 4.8.8.
Evaluate \(\int x^3+2x+3\, dx\)
Evaluate \(\int t^{-2/3} - \frac{5}{t}\, dt\)
Evaluate \(\int \frac{3}{4x^2+1}\, dx\)
\(\displaystyle \frac14 x^4 + x^2 + 3x + C\)
\(\displaystyle 3t^{1/3} - 5\ln |t| + C\)
\(\displaystyle \frac32\tan^{-1} (2x) + C\)
Subsection 4.8.2 Initial value problems and differential equations
Definition 4.8.9. Differential equation.
A differential equation, loosely, is an equation involving the derivative of an unknown function.
Example 4.8.10.
\(\displaystyle\frac{dy}{dx} = x^2\) is a differential equation. The antiderivative \(y=\frac{1}{3}x^3+C\) is the general solution.
Adding a specific value, for example, \(y(0)=5\text{,}\) makes the problem into an initial value problem, for which \(y=\frac{1}{3}x^3+5\) is the particular solution.
Example 4.8.11.
A rocket lifts off with constant acceleration of \(20 \text{m}/\text{sec}^2\text{.}\) How fast will it be going 1 minute later?
Example 4.8.12.
Illinois requires motorcycles to be able to stop from \(44 \text{feet}/\text{sec}\) to 0 in 45 feet. What constant deceleration is required to achieve that stopping distance?
\(\frac{d^2s}{dt^2} = -k\) for an unknown \(k\)
Integrating: \(\frac{ds}{dt} = -kt+C\text{;}\) since \(v(0)=44\text{,}\) we have
Integrating: \(s = -\frac{1}{2}kt^2 + 44t + C\text{;}\) since \(s(0)=0\text{,}\) we have
The time to stop is the solution of \(-kt+44=0\text{,}\) so \(t_{\text{stop}}=\frac{44}{k}\text{.}\)
So, \(45 = -\frac{1}{2}k\left(\frac{44}{k}\right)^2 + 44 \left(\frac{44}{k}\right)\) yields \(k=\frac{90}{44^2}\text{.}\)