Just like differentiation has rules, integration has its own set of basic rules. These rules allow you to find antiderivatives of common functions without starting from scratch every time.
🎯 In this section you will learn
- The Power Rule for Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
- The Constant Multiple Rule: $\int c \cdot f(x) dx = c \int f(x) dx$
- The Sum/Difference Rule: $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$
- Special integrals: $\int e^x dx$, $\int \frac{1}{x} dx$, $\int \sin x dx$, $\int \cos x dx$
📋 The Basic Integration Rules (Reference Table)
| Rule | Formula | Example |
|---|
| Power Rule (n ≠ -1) | $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ | $\int x^4 dx = \frac{x^5}{5} + C$ |
| Constant Multiple Rule | $\int c \cdot f(x) dx = c \int f(x) dx$ | $\int 5x^2 dx = 5 \cdot \frac{x^3}{3} + C$ |
| Sum Rule | $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$ | $\int (x^2 + \sin x) dx = \frac{x^3}{3} - \cos x + C$ |
| Difference Rule | $\int [f(x) - g(x)] dx = \int f(x) dx - \int g(x) dx$ | $\int (x^3 - e^x) dx = \frac{x^4}{4} - e^x + C$ |
| Exponential | $\int e^x dx = e^x + C$ | $\int 3e^x dx = 3e^x + C$ |
| Natural Log | $\int \frac{1}{x} dx = \ln|x| + C$ | $\int \frac{5}{x} dx = 5\ln|x| + C$ |
| Sine | $\int \sin x dx = -\cos x + C$ | $\int 2\sin x dx = -2\cos x + C$ |
| Cosine | $\int \cos x dx = \sin x + C$ | $\int 3\cos x dx = 3\sin x + C$ |
📌 Power Rule in Depth
$$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1) $$
- $\int x^3 dx = \frac{x^4}{4} + C$
- $\int x^7 dx = \frac{x^8}{8} + C$
- $\int x^{10} dx = \frac{x^{11}}{11} + C$
- $\int x^{-3} dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$
- $\int x^{-2} dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$
- $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C$
📌 Constant Multiple and Sum/Difference Rules
Find $\int (3x^2 - 4x + 5) dx$.
①
Break into separate integrals
$\int 3x^2 dx - \int 4x dx + \int 5 dx$
②
Apply constant multiple rule
$3\int x^2 dx - 4\int x dx + 5\int 1 dx$
③
Apply power rule
$3 \cdot \frac{x^3}{3} - 4 \cdot \frac{x^2}{2} + 5x + C = x^3 - 2x^2 + 5x + C$
📌 Special Integrals
- $\int e^x dx = e^x + C$
- $\int 5e^x dx = 5e^x + C$
- $\int \frac{1}{x} dx = \ln|x| + C$
- $\int \frac{3}{x} dx = 3\ln|x| + C$
- $\int \sin x dx = -\cos x + C$
- $\int \cos x dx = \sin x + C$
- $\int (2\sin x + 3\cos x) dx = -2\cos x + 3\sin x + C$
Find $\int (x^2 + e^x + \frac{1}{x} + \sin x) dx$.
- $\int x^2 dx = \frac{x^3}{3}$
- $\int e^x dx = e^x$
- $\int \frac{1}{x} dx = \ln|x|$
- $\int \sin x dx = -\cos x$
- $\int (x^2 + e^x + \frac{1}{x} + \sin x) dx = \frac{x^3}{3} + e^x + \ln|x| - \cos x + C$
⚠️ Common Mistakes
- Forgetting the $+C$: Indefinite integrals always include the constant of integration.
- Adding 1 to the exponent incorrectly: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, not $\frac{x^{n+1}}{n} + C$.
- Missing absolute value: $\int \frac{1}{x} dx = \ln|x| + C$, not $\ln x + C$ (the absolute value extends the domain).
- Sign errors with $\sin$ and $\cos$: $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$.
🔍 Key Takeaways
- Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (except $n = -1$).
- Constant multiple: $\int c f(x) dx = c \int f(x) dx$.
- Sum/Difference: $\int (f \pm g) = \int f \pm \int g$.
- Memorize special integrals: $e^x$, $\frac{1}{x}$, $\sin x$, $\cos x$.
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