Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
Introduced| English | Meaning |
|---|---|
| indefinite integral/ɪnˈdefɪnət ˈɪntɪɡrəl/ | The family of antiderivatives of a function, usually expressed with an arbitrary constant. |
| constant of integration/ˈkɒnstənt ɒv ˌɪntɪˈɡreɪʃn/ | An arbitrary constant included in an indefinite integral to represent the family of antiderivatives. |
Running differentiation backward
- FTC Part 2 needs an antiderivative — so let's learn to find them.
- An antiderivative of $f$ is any function whose derivative is $f$; the whole family is the indefinite integral 不定积分.
- $\displaystyle\int f(x)\,dx$ (no limits) means "the general antiderivative of $f$."
- Every derivative rule, read backward, becomes an antiderivative rule.
The power rule for integration
- Reverse the Power Rule: add one to the exponent, then divide by the new exponent.
-
$$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\qquad(n\neq-1)$$
- Check: differentiating $\frac{x^{n+1}}{n+1}$ gives $x^n$. ✓
- (The case $n=-1$ is special: $\int x^{-1}\,dx=\ln|x|+C$.)
Antiderivatives differ by a constant: vertical shifts with the same slopes
What is $\displaystyle\int x^3\,dx$?
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Add one to the exponent, divide by it: $\tfrac{x^4}{4}+C$.
The power rule fails for $n=-1$. What is $\displaystyle\int x^{-1}\,dx$?
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$\int\tfrac1x\,dx=\ln|x|+C$.
Never forget $+C$
- Since the derivative of a constant is $0$, any constant can be added — so the antiderivative is a family.
- Always append the constant of integration 积分常数 $+C$ to an indefinite integral.
- Without limits there's no way to pin down which constant, so $+C$ stays.
- (In a definite integral the $+C$ cancels — that's the difference.)
A family shifted by +C
y = ax³ + d
All antiderivatives of $f$ differ only by a vertical shift $+C$ — same slope everywhere.
An indefinite integral should include the constant of integration $+C$.
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The antiderivative is a family; $+C$ is part of the answer.
The standard antiderivatives
- Reverse the elementary derivatives you know:
- $\displaystyle\int\cos x\,dx=\sin x+C$; $\;\displaystyle\int\sin x\,dx=-\cos x+C$.
- $\displaystyle\int e^x\,dx=e^x+C$; $\;\displaystyle\int\frac1x\,dx=\ln|x|+C$.
- Combine with linearity to integrate any polynomial or sum term by term.
What is $\displaystyle\int \sin x\,dx$?
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$\int\sin x\,dx=-\cos x+C$ (mind the minus).
Find $\displaystyle\int (6x^2-4x+5)\,dx$.
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Reverse power rule term by term, plus $C$.
Select all correct antiderivatives.
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$\int\cos x\,dx=\sin x+C$ (positive); the last is wrong.
Two staples: always add $+C$ to an indefinite integral (it's part of the answer), and the power rule fails for $n=-1$ — $\int x^{-1}\,dx$ is $\ln|x|+C$, not $\frac{x^0}{0}$. Also mind the sign on $\int\sin x\,dx=-\cos x+C$ (a minus, mirroring $\cos$'s derivative).
Find $\displaystyle\int (6x^2 - 4x + 5)\,dx$.
- Term by term (reverse power rule): $6\cdot\dfrac{x^3}{3}=2x^3$; $\;-4\cdot\dfrac{x^2}{2}=-2x^2$; $\;5x$.
- $\displaystyle\int (6x^2-4x+5)\,dx = 2x^3-2x^2+5x+C$.
- Check by differentiating: $6x^2-4x+5$. ✓
An indefinite integral $\int f\,dx$ is the general antiderivative of $f$. The power rule for integration: $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq-1$ (and $\int x^{-1}\,dx=\ln|x|+C$). Always add the constant of integration $+C$, and integrate sums term by term.