Indefinite Integration — The Power Rule

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Integration is the reverse of differentiation: The antiderivative of f(x) is a function F(x) such that F'(x) = f(x). Write: \int f(x) dx = F(x) + C The +C is the constant of integration — it represents any constant that disappears when differentiating.

The power rule for integration: \int xⁿ dx = (xⁿ⁺¹)(n+1) + C for n ≠ −1 Add 1 to the power, then divide by the new power. Exception: \int x^-1 dx = \int (1)/(x) dx = ln|x| + C

Do NOT forget +C: An indefinite integral ALWAYS includes +C. Why? Because when you differentiate any constant, you get 0. So the original function could have ANY constant added to it. Mi ing +C in IB exams loses a mark automatically — do not forget it.

Standard integrals to know: • \int xⁿ dx = (xⁿ⁺¹)(n+1) + C • \int (1)/(x) dx = ln|x| + C • \int e^x dx = e^x + C • \int e^kx dx = (1)/(k)e^kx + C • \int sin x dx = -cos x + C • \int cos x dx = sin x + C

Finding a specific antiderivative: To find C, you need an initial condition — a known point (x₀, y₀) on the curve. Substitute into F(x) + C = y₀ and solve for C. This gives the particular solution (one specific curve, not the family of curves).

Integration of exponential functions: \int e^{ax + b} dx = (1)/(a)e^{ax + b} + C The chain rule in reverse: divide by the coefficient of x inside the exponent.

Verify by differentiating: You can ALWAYS verify an integral by differentiating your answer. If you differentiate (5)/(3)e^3x + C, you get (5)/(3) · 3 · e^3x = 5e^3x. Correct! Checking by differentiation is fast and catches most integration error.

Integration by substitution: Substitution (u-substitution) is used when the integrand contains a composite function. Look for: an inner function g(x) whose derivative g'(x) also appears in the integrand. Set u = g(x), find du = g'(x) dx, substitute to simplify, integrate in u, substitute back.

Integration of ln form: If the integrand has the form (f'(x))(f(x)), the integral is ln|f(x)| + C. This is because (d)/(dx)[ln|f(x)|] = (f'(x))(f(x)).'

Recognising the pattern: Before attempting substitution, ask: • Is there a function inside a power? → substitution • Is the numerator the derivative of the denominator? → ln form • Is the exponent a linear function of x? → reverse chain rule For IB AI SL, most integration questions involve one of these three pattern.

Two constants in motion problems: When integrating twice (acceleration → velocity → position), you get two constants C₁ and C₂. You need TWO initial conditions to find both constants — usually v(0) and s(0) are given. Solve them in order: first find C₁ from v, then find C₂ from .