Integration Rules & Formulas — The Complete Reference

Reverse Power Rule

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1.
• Exceptions: ∫1/x dx = ln|x|+C (the rule breaks at n=−1, which is why this special case exists).
• Examples: ∫x⁴dx = x⁵/5+C, ∫√x dx = (2/3)x^(3/2)+C, ∫x⁻³dx = −1/(2x²)+C.

Linearity of Integration

• ∫[af(x)+bg(x)]dx = a∫f(x)dx + b∫g(x)dx.
• You can split a sum and factor out constants. This is why polynomials integrate term-by-term.

Exponential and Log Integrals

• ∫eˣ dx = eˣ+C
• ∫eˢˣ dx = eˢˣ/s+C (s≠0)
• ∫aˣ dx = aˣ/ln(a)+C
• ∫1/x dx = ln|x|+C

Trigonometric Integrals

• ∫sin x dx = −cos x+C
• ∫cos x dx = sin x+C
• ∫sec²x dx = tan x+C
• ∫csc²x dx = −cot x+C
• ∫sec x tan x dx = sec x+C

Inverse Trig Integrals

• ∫1/√(1−x²) dx = arcsin x+C
• ∫1/(1+x²) dx = arctan x+C
• These arise constantly in applications and are derived from differentiating arcsin and arctan.

How to Approach Any Integral

Unlike differentiation — where every elementary function has a computable derivative and a systematic set of rules covers all cases — integration requires strategy. Before applying any technique, always ask: does this integrand match a known formula directly? If yes, use the formula. If the integrand is a composite (something inside something), try u-substitution. If it is a product of unrelated functions, try integration by parts. If it is a rational function, try partial fractions. If it involves √(a²−x²) or √(a²+x²), try trigonometric substitution.

Pattern recognition is the core skill of integration. It comes from seeing many examples, not from memorising algorithms. The more integrals you compute, the faster you recognise which technique applies.

Power Rule — Every Case

∫xⁿ dx = xⁿ⁺¹/(n+1) + C, for n ≠ −1. This covers integers, fractions, and negative exponents — any power except −1. The exception n = −1 gives ∫x⁻¹ dx = ∫(1/x) dx = ln|x| + C. The absolute value in ln|x| is necessary: the integral is defined for both positive and negative x (separately), and ln requires a positive argument.

• ∫x⁷ dx = x⁸/8 + C
• ∫x^(−3) dx = x^(−2)/(−2) + C = −1/(2x²) + C
• ∫x^(1/2) dx = x^(3/2)/(3/2) + C = (2/3)x^(3/2) + C = (2/3)√(x³) + C
• ∫x^(−1/2) dx = x^(1/2)/(1/2) + C = 2√x + C
• ∫(1/x⁴) dx = ∫x⁻⁴ dx = x⁻³/(−3) + C = −1/(3x³) + C

Linearity — Splitting Integrals

The linearity of integration means you can split sums and factor out constants: ∫[af(x) + bg(x)]dx = a∫f(x)dx + b∫g(x)dx. This makes polynomial integration completely mechanical — just apply the power rule to each term. ∫(5x⁴ − 3x² + 7x − 2)dx = x⁵ − x³ + (7/2)x² − 2x + C. Never forget to add a single +C at the end, not one per term.

Exponential and Logarithmic Integrals — Derived

∫eˣ dx = eˣ + C. This follows from the fact that eˣ is its own derivative. For ∫aˣ dx: since aˣ = e^(x ln a), differentiating gives ln(a)·e^(x ln a) = ln(a)·aˣ. Therefore ∫aˣ dx = aˣ/ln(a) + C. Note this requires a > 0 and a ≠ 1.

For ∫1/x dx = ln|x| + C: verify by differentiating — d/dx[ln|x|] = 1/x for both x > 0 and x < 0. The absolute value is essential because the natural log is only defined for positive arguments, but 1/x is integrable on (−∞, 0) as well.

Recognising Reverse Chain Rule Patterns

Many integrals are best recognised as the reverse of a chain rule derivative rather than a u-substitution computation. If you see ∫2x(x²+1)⁵ dx, notice that 2x is the derivative of x²+1 — this is ∫g'(x)·[g(x)]⁵ dx, which integrates to [g(x)]⁶/6 + C = (x²+1)⁶/6 + C. Similarly, ∫(cos x)·e^(sin x) dx: cos x is the derivative of sin x, so this integrates to e^(sin x) + C.

Training your eye to spot these patterns dramatically speeds up integration. The pattern is: ∫f'(g(x))·g'(x)dx = f(g(x)) + C. When you see a function and its derivative multiplied together, the integral is just the antiderivative of the outer function evaluated at the inner function.

Complete Reference Table

• ∫xⁿ dx = xⁿ⁺¹/(n+1) + C  (n ≠ −1)
• ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C
• ∫aˣ dx = aˣ/ln(a) + C
• ∫sin x dx = −cos x + C
• ∫cos x dx = sin x + C
• ∫sec²x dx = tan x + C
• ∫csc²x dx = −cot x + C
• ∫sec x tan x dx = sec x + C
• ∫csc x cot x dx = −csc x + C
• ∫1/√(1−x²) dx = arcsin x + C
• ∫1/(1+x²) dx = arctan x + C
• ∫tan x dx = ln|sec x| + C
• ∫sec x dx = ln|sec x + tan x| + C