Lesson 16 Antiderivatives

16.1 Lecture Content

16.2 Lecture Notes

16.2.1 Basic Antiderivatives and Rules

Antiderivatives are the reverse of derivatives. The antiderivative of a function $f(x)$ is a function $F(x)$ such that:

\[ F'(x) = f(x) \]

The indefinite integral of $f(x)$ is:

\[ \int f(x) \, dx = F(x) + C \]

Where $C$ is the constant of integration.

16.2.1.1 Rules of Antidifferentiation

Sum Rule:
\[ \int (f(x) + g(x))\,dx = \int f(x)\,dx + \int g(x)\,dx \]
Constant Multiple Rule:
\[ \int c \cdot f(x)\,dx = c \cdot \int f(x)\,dx \]
Power Rule: \[ \int x^n\,dx = \frac{x^{n+1}}{n+1} + C \text{ for } n \ne -1 \]
Exponential Rule: \[ \int e^x \,dx = e^x + C \]
Reciprocal Rule: \[ \int \frac{1}{x} \,dx = \ln|x| + C \]

16.2.1.2 Common Antiderivatives

Function $f(x)$	Antiderivative $\int f(x)\,dx$
$x^n$ (for $n \ne -1$)	$\frac{x^{n+1}}{n+1} + C$
$\frac{1}{x}$	$\ln\|x\| + C$
$e^x$	$e^x + C$
$a^x$	$\frac{a^x}{\ln a} + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\sec x \tan x$	$\sec x + C$

16.2.2 Integration by Substitution

Substitution is used when the integrand contains a composite function.

16.2.2.1 Steps:

Choose a substitution:
Let $u = g(x)$, where $g(x)$ is an inner function.
Differentiate $u$:
Compute $du = g'(x)\,dx$.
Rewrite the integral:
Replace all $x$-terms with $u$-terms.
Integrate in terms of $u$.
Substitute back:
Replace $u$ with the original expression.

Example:

\[ \int 2x \cos(x^2) \, dx \]

Let $u = x^2$, then $du = 2x \, dx$.
So the integral becomes:

\[ \int \cos(u) \, du = \sin(u) + C = \sin(x^2) + C \]

16.2.3 Integration by Parts

Use this method when the integrand is a product of two functions.

16.2.3.1 Formula:

\[ \int u\,dv = uv - \int v\,du \]

16.2.3.2 Steps:

Choose $u$ and $dv$ from the integrand.
- Use LIPET (Logarithmic, Inverse trig, Polynomial, Exponential, Trig) as a guide for choosing $u$.
Differentiate $u$ to get $du$, and integrate $dv$ to get $v$.
Apply the formula:
\[ \int u\,dv = uv - \int v\,du \]
Simplify and integrate again if necessary.

Example:

\[ \int x e^x \, dx \]

Let $u = x$, $dv = e^x dx$
Then $du = dx$, $v = e^x$

\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C \]

16.3 Examples

16.3.1 Antiderivatives

Video: Solutions

Consider the function $f(x) = 6x^{10} + 2x^6 - 4x^4 - 4$. Find an antiderivative of $f(x)$.
Evaluate the definite integral $\int_5^8 \frac{1}{x^2} \, dx$.
Find the indefinite integral $\int \left(\frac{4}{x^5} + 3x + 5 \right) \, dx$.
Compute the indefinite integral $\int (x - 4)(x + 4) \, dx$.
Find the indefinite integral $\int \left(-5x^4 + \frac{3}{x} - \frac{3}{x^2} - 4\sqrt{x} \right) \, dx$.
Consider the function $f(t) = 10 \sec^2(t) - 3 t^2$. Let $F(t)$ be the antiderivative of $f(t)$ such that $F(0) = 0$. Find $F(3)$.
Find a function $f(x)$ such that $f'(x) = 4 e^x + 3x$ and $f(0) = -5$.

16.3.2 U-Substitution

Video: Solutions

Evaluate the integral $\int x^6 (x^7 - 11)^{50} \, dx$.
Evaluate the indefinite integral $\int x^3 \sqrt{11 + x^4} \, dx$.
Evaluate the indefinite integral $\int \frac{x^5}{x^6 + 2} \, dx$.
Evaluate the indefinite integral $\int \frac{6}{x \ln(9x)} \, dx$.
Evaluate the indefinite integral $\int 3 e^{3x} \sin\bigl(e^{3x}\bigr) \, dx$.
Evaluate the indefinite integral $\int \frac{x + 2}{x^2 + 4x + 5} \, dx$.
Evaluate the indefinite integral $\int x^6 e^{x^7} \, dx$.

16.3.3 Integration by Parts

Video: Solutions

Use integration by parts to evaluate the definite integral $\int_1^e 3t^2 \ln t \, dt$.
Evaluate the indefinite integral $\int 6x e^x \, dx$.
Evaluate the indefinite integral $\int 6x e^{4x} \, dx$.
Use integration by parts to evaluate the integral $\int 8z \cos(4z) \, dz$.
Use integration by parts to evaluate the integral $\int x^6 e^x \, dx$.

16.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

Evaluate the following integrals:
1. \[\int (x + x^3) \, dx\]
2. \[\int (\sin u + \cos u) \, du\]
3. \[\int \sqrt[4]{x} + \sin x \, dx\]
4. \[\int \frac{1}{x} + \frac{2}{x^2} \, dx\]
Evaluate the following integrals using the substitution method:
1. \[\int (2x+5)^{-3} \, dx\]
2. \[\int \sqrt{4x-5} \, dx\]
3. \[\int \sin(4u+6) \, du\]
4. \[\int x e^{-x^2+1} \, dx\]
Evaluate the following integrals using integration by parts:
1. \[\int 2x e^{x} \, dx\]
2. \[\int (x^2 - 1) 3^{-x} \, dx\]
3. \[\int x^{2} \ln x \, dx\]
4. \[\int x^{3} (x+1)^{10} \, dx\]
(Applied) The marginal cost of producing the $x$th box of CDs is given by
\[10 - \frac{x}{(x^{2} + 1)^{2}}.\]
The total cost to produce 2 boxes is $1,000. Find the total cost function $C(x)$.
(Applied) The number of housing starts in the United States can be approximated by
\[n(t) = \frac{1}{12} \left(1.1 + 1.2 e^{-0.08 t} \right) \text{ million homes per month } (t \geq 0)\]
where $t$ is time in months from the start of 2006. Find an expression for the total number $N(t)$ of housing starts in the US from January 2006 to time $t$.

16.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

Evaluate the following integrals:
1. \[\int \left(\frac{1}{v^2} + \frac{2}{v}\right) dv\]
2. \[\int \frac{|u|}{u} + \sec^{2} u \, du\]
Evaluate the following integrals using the substitution method:
\[\int \left( 2x e^{x^{2} + 4} + \frac{5}{2x + 8} \right) dx\]
Evaluate the following integrals using integration by parts:
\[\int x \ln(2x) \, dx\]
The marginal cost of producing the $x$th box of Zip disks is
\[10 + \frac{x^{2}}{100,000}\]
and the fixed cost is $100,000. Find the cost function $C(x)$.

16.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an $X$ in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill	D	CON	COM
Evaluate an integral using basic techniques.
Evaluate an integral using the substitution method.
Evaluate an integral using integration by parts.
Solve applied problems using antiderivatives.

Function \(f(x)\)	Antiderivative \(\int f(x)\,dx\)
\(x^n\) (for \(n \ne -1\))	\(\frac{x^{n+1}}{n+1} + C\)
\(\frac{1}{x}\)	\(\ln\|x\| + C\)
\(e^x\)	\(e^x + C\)
\(a^x\)	\(\frac{a^x}{\ln a} + C\)
\(\sin x\)	\(-\cos x + C\)
\(\cos x\)	\(\sin x + C\)
\(\sec^2 x\)	\(\tan x + C\)
\(\sec x \tan x\)	\(\sec x + C\)