Integration is the reverse of differentiation. It is used to find areas, volumes, and solve differential equations — one of the most powerful tools in mathematics.
∫xⁿ dx = xⁿ⁺¹/(n+1) + c (n ≠ -1). ∫eˣ dx = eˣ + c. ∫1/x dx = ln|x| + c. ∫sinx dx = -cosx + c. ∫cosx dx = sinx + c. Reverse chain rule: ∫(ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / (a(n+1)) + c. Always add constant c for indefinite integrals.
Definite integral ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) - F(a). Area under curve between x = a and x = b: ∫ₐᵇ y dx. If curve below x-axis: integral is negative, take absolute value. Area between two curves: ∫ₐᵇ (f(x) - g(x)) dx where f(x) > g(x). Split integral if curves cross within the range.
Substitution: replace u = g(x), change dx to du. Integration by parts: ∫u dv = uv - ∫v du (LIATE rule for choosing u). Partial fractions: decompose rational function, then integrate term by term. Example: ∫1/(x(x+1)) dx = ∫(1/x - 1/(x+1)) dx = ln|x| - ln|x+1| + c. Volume of revolution about x-axis: V = π∫ₐᵇ y² dx. About y-axis: V = π∫ₐᵇ x² dy.
When you differentiate a constant, you get zero. So when you reverse the process (integrate), there could have been any constant term in the original function. For example: d/dx(x² + 5) = 2x and d/dx(x² - 3) = 2x — both give the same derivative. So ∫2x dx = x² + c where c could be any constant. The +c represents all possible antiderivatives. For definite integrals, the constant cancels: [F(x) + c]ₐᵇ = (F(b)+c) - (F(a)+c) = F(b) - F(a), so we do not need to write it.
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