Integration and its Applications — Chapter Summary

If d(F(x))/dx = f(x)

If d(F(x))/dx = f(x), then ∫f(x)dx = F(x) + C, where C is an arbitrary constant. The integral represents a family of curves that differ by a constant

Key formulae include

Key formulae include: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n≠-1), ∫1/x dx = log|x| + C, ∫eˣ dx = eˣ + C, ∫aˣ dx = aˣ/log a + C. These serve as building blocks for

Method where we substitute a part

Method where we substitute a part of the integrand with a new variable to simplify the integral. Rule: ∫f(g(x))g'(x)dx = ∫f(t)dt where g(x) = t. Commo

Technique for integrating rational functions P(x)/Q(x)

Technique for integrating rational functions P(x)/Q(x) by decomposing them into simpler fractions. For example, 1/((x-1)(x+3)) = A/(x-1) + B/(x+3). Ea

Used for integrating products of functions

Used for integrating products of functions using the formula: ∫f(x)g(x)dx = f(x)∫g(x)dx - ∫[f'(x)∫g(x)dx]dx. Choice of first function follows ILATE ru