Integration Rules allow systematic evaluation of integrals by transforming complex functions into standard forms using linearity, substitution, and parts methods.
## Core concept
Integration is the reverse of differentiation. While differentiation breaks functions down, integration builds them back up. Integration rules let you handle sums, products, and composite functions without memorising every integral separately.
The fundamental principle is that integration is linear: $$\int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx$$
where $a, b$ are constants and $f(x), g(x)$ are integrable functions.
## Main integration rules
### Rule 1: Power Rule $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$
Example: $\int x^3 \, dx = \frac{x^4}{4} + C$
### Rule 2: Constant Multiple Rule $$\int k \cdot f(x) \, dx = k \int f(x) \, dx$$
A constant can be pulled outside the integral.
### Rule 3: Sum/Difference Rule $$\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$$
Integrate each term separately, then combine.
### Rule 4: Substitution (Change of Variable) When the integrand contains a composite function: - Let $u = g(x)$, so $du = g'(x) \, dx$ - Rewrite the integral in terms of $u$ - Integrate - Substitute back
Example: $\int 2x(x^2 + 1)^5 \, dx$
Let $u = x^2 + 1$, then $du = 2x \, dx$
$\int u^5 \, du = \frac{u^6}{6} + C = \frac{(x^2 + 1)^6}{6} + C$
### Rule 5: Integration by Parts $$\int u \, dv = uv - \int v \, du$$
Choose $u$ and $dv$ using the ILATE priority (Inverse trig, Logarithm, Algebraic, Trigonometric, Exponential).
Example: $\int x e^x \, dx$
Let $u = x$ (algebraic), $dv = e^x \, dx$
Then $du = dx$, $v = e^x$
$\int x e^x \, dx = x e^x - \int e^x \, dx = xe^x - e^x + C = e^x(x-1) + C$
## Common exam applications
- Breaking down polynomials: Use sum rule on $\int (3x^2 + 2x - 5) \, dx$
- Rational exponents: Convert to power form: $\int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{2x^{3/2}}{3} + C$
- Exponential and logarithmic integrals: Substitution is often needed
- Area under curves: Definite integrals (covered separately) apply these rules with limits
## Common mistakes
- Forgetting the constant of integration (+C) — Always include it in indefinite integrals
- Wrong power rule exponent: $\int x^2 \, dx \neq x^3 + C$ (the denominator is $n+1 = 3$)
- Not simplifying before integrating: Expand or simplify the integrand first
- Incorrect substitution: Ensure $du$ matches your differential; if not, scale appropriately
- Integration by parts confusion: Choosing $u$ wrongly (use ILATE) or making sign errors in $uv - \int v \, du$
Exam tip: Most CA Foundation questions test power rule, sum rule, and basic substitution. Master these first, then practise integration by parts on products of polynomial and exponential/trig functions.