Antiderivatives and Indefinite Integrals
Definition of an Antiderivative
A function $F$ is called an antiderivative of $f$ on an interval $I$ if $F^{\prime}(x)=f(x)$ for all $x$ on $I$.
Representation Antiderivatives
If $F$ is an antiderivative of $f$ on an interval $I$, then $F(x)+C$ represents the most general antiderivative of $f$ on $I$, where $C$ is a constant.
Example 1
- Find an antiderivative for each of the following functions.
- $f(x)=3 x^2$
- $g(x)=\cos x+3$
▶️Answer/Explanation
Solution
a. $F(x)=x^3+C$ Derivative of $x^3$ is $3 x^2$.
b. $G(x)=\sin x+3 x+C$ Derivative of $\sin x$ is $\cos x$, and derivative of $3 x$ is 3 .
Definition of Indefinite Integral
The set of all antiderivatives of $f$ is the indefinite integral of $f$ with respect to $x$ denoted by
$
\int f(x) d x
$
Thus $\int f(x) d x=F(x)+C$ means $F^{\prime}(x)=f(x)$.
Example 2
- Find the general solution of $F^{\prime}(x)=\sec ^2 x$.
▶️Answer/Explanation
Solution
$F(x)=\int \sec ^2 x=\tan x+C \quad$ Derivative of $\tan x$ is $\sec ^2 x$.
Table of Indefinite Integrals
- $\int k d x=k x+C \quad \quad \int k f(x) d x=k \int f(x) d x $
- $\int[f(x) \pm g(x)] d x=\int f(x) d x \pm \int g(x) d x $
- $\int x^n d x=\frac{x^{n+1}}{n+1}+C, n \neq-1 \quad \text { Power Rule } $
- $\int e^x d x=e^x+C $
- $\int \sin x d x=-\cos x+C \quad \quad \int \cos x d x=\sin x+C $
- $\int \sec ^2 x d x=\tan x+C \quad \quad \int \csc ^2 x d x=-\cot x+C $
- $\int \sec x \tan x d x=\sec x+C \quad \quad \int \csc ^2 x \cot x d x=-\csc x+C$
Example 3
- Find the antiderivative of $x^3-3 x+2$.
▶️Answer/Explanation
Solution
$
\begin{aligned}
& \int\left(x^3-3 x+2\right) d x=\int x^3 d x-\int 3 x d x+\int 2 d x \\
& =\frac{x^{3+1}}{3+1}-3 \cdot \frac{x^{1+1}}{1+1}+2 x+C \quad \text { Power Rule } \\
& =\frac{x^4}{4}-\frac{3 x^2}{2}+2 x+C \quad \text { Answer } \\
& \text { Check: } \frac{d}{d x}\left(\frac{x^4}{4}-\frac{3 x^2}{2}+2 x+C\right)=\frac{1}{4} \cdot 4 x^3-\frac{3}{2} \cdot 2 x+2 \cdot 1+0 \\
& =x^3-3 x+2 \quad \checkmark
\end{aligned}
$
Example 4
- Find the general indefinite integral
$
\int(\sqrt{x}-\sec x \tan x) d x \text {. }
$
▶️Answer/Explanation
Solution
$
\begin{aligned}
& \int(\sqrt{x}-\sec x \tan x) d x=\int \sqrt{x} d x-\int \sec x \tan x d x \\
& =\frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\sec x+C=\frac{2}{3} x^{\frac{3}{2}}-\sec x+C
\end{aligned}
$
$
\text { Check: } \begin{aligned}
& \frac{d}{d x}\left(\frac{2}{3} x^{\frac{3}{2}}-\sec x+C\right)=\frac{2}{3} \cdot \frac{3}{2} x^{\frac{3}{2}-1}-\sec x \tan x+0 \\
= & x^{\frac{1}{2}}-\sec x \tan x=\sqrt{x}-\sec x \tan x \quad \sqrt{ }
\end{aligned}
$
Example5
- If $\frac{d y}{d x}=3 x^2-1$, and if $y=-1$ when $x=1$, then $y=$
(A) $x^3-x+1$
(B) $x^3-x-1$
(C) $-x^3+x-1$
(D) $-x^3+1$
▶️Answer/Explanation
Ans:B
- Which of the following is the antiderivative of $f(x)=\tan x$ ?
(A) $\sec x+\tan x+C$
(B) $\csc x+\cot x+C$
(C) $\ln |\csc x|+C$
(D) $-\ln |\cos x|+C$
▶️Answer/Explanation
Ans:D
- A curve has a slope of $-x+2$ at each point $(x, y)$ on the curve. Which of the following is an equation for this curve if it passes through the point $(2,1)$ ?
(A) $\frac{1}{2} x^2-2 x-4$
(B) $2 x^2+x-8$
(C) $-\frac{1}{2} x^2+2 x-1$
(D) $x^2-2 x+1$
▶️Answer/Explanation
Ans:C