Indefinite Integrals and Substitution

Indefinite Integrals

  • The indefinite integral of a function $f(x)$ is the family of all antiderivatives, denoted as $\int f(x) , dx = F(x) + C$, where $C$ is the constant of integration.
  • Basic integrals are derived from differentiation rules and must be memorised, e.g.:
    • $\int e^{kx} , dx = \frac{1}{k} e^{kx} + C$
    • $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
    • $\int \sin x , dx = -\cos x + C$
    • $\int \sec^2 x , dx = \tan x + C$
    • $\int \sec x \tan x , dx = \sec x + C$

Example:
$$ \int (3e^{4x} - 5e^{-6x}) , dx = \frac{3}{4} e^{4x} + \frac{5}{6} e^{-6x} + C $$

Substitution Method

  • The substitution method (or $u$-substitution) reverses the chain rule:
    $$\int f(g(x)) , g'(x) , dx = \int f(u) , du, \quad u = g(x), \quad du = g'(x) , dx$$
  • Steps:
    1. Choose $u = g(x)$ such that $du = g'(x) , dx$ appears in the integrand.
    2. Rewrite the integral entirely in terms of $u$ and $du$.
    3. Integrate with respect to $u$.
    4. Substitute back $u = g(x)$.

Example 1:
Find $\int 3x^2 (x^3 - 2)^2 , dx$.

Let $u = x^3 - 2$, so $du = 3x^2 , dx$. Then
$$ \int 3x^2 (x^3 - 2)^2 , dx = \int u^2 , du = \frac{u^3}{3} + C = \frac{(x^3 - 2)^3}{3} + C $$

Example 2:
Find $\int \frac{\sec^2 x}{(1 + \tan x)^3} , dx$.

Let $u = 1 + \tan x$, so $du = \sec^2 x , dx$. Then
$$ \int \frac{\sec^2 x}{(1 + \tan x)^3} , dx = \int u^{-3} , du = \frac{u^{-2}}{-2} + C = -\frac{1}{2(1 + \tan x)^2} + C $$

Integration Tips

  • For integrals like $\int \frac{x^3 + 2x^2 + 5x + 6}{x+2} , dx$, perform polynomial long division first.
  • For trigonometric integrals, use identities such as $\sin^2 x + \cos^2 x = 1$ to simplify.
  • Substitution often works when the integrand contains a function and its derivative (e.g., $\frac{\ln x}{x}$, $e^{x^2} \cdot 2x$, $\tan x \sec^2 x$).