Indefinite Integration – Core Formulas & Techniques

These question-and-answer flashcards cover basic antiderivatives, the classic “10 sister” formulas, trigonometric integrals, integration by parts, shortcuts, partial fractions, and standard substitutions for irrational algebraic expressions.

How do you integrate ∫ cos x dx?

∫ cos x dx = sin x + C

What is the result of ∫ sin x dx?

∫ sin x dx = –cos x + C

Which basic integral equals tan x + C?

∫ sec² x dx = tan x + C

What is the antiderivative of sec x tan x?

∫ sec x tan x dx = sec x + C

Give the integral of eˣ.

∫ eˣ dx = eˣ + C

How is ∫ aˣ dx expressed (a > 0, a ≠ 1)?

∫ aˣ dx = aˣ / ln a + C

What logarithmic form is produced by ∫ sec x dx?

∫ sec x dx = ln|sec x + tan x| + C

State the integral of ∫ cosec x dx.

∫ cosec x dx = –ln|cosec x + cot x| + C

Which integral gives ln|sec x| + C?

∫ tan x dx = ln|sec x| + C

What is the antiderivative of cot x?

∫ cot x dx = ln|sin x| + C

Give the first of the “10 sister” formulas: ∫ dx ⁄ (x² + a²).

∫ dx ⁄ (x² + a²) = (1/a) tan⁻¹(x/a) + C

Provide the sister formula for ∫ dx ⁄ (x² – a²).

∫ dx ⁄ (x² – a²) = (1/2a) ln|(x – a)/(x + a)| + C

What is ∫ dx ⁄ √(x² + a²)?

∫ dx ⁄ √(x² + a²) = ln|x + √(x² + a²)| + C

Give the integral of 1 ⁄ √(x² – a²).

∫ dx ⁄ √(x² – a²) = ln|x + √(x² – a²)| + C

How do you integrate 1 ⁄ √(a² – x²)?

∫ dx ⁄ √(a² – x²) = sin⁻¹(x/a) + C

What is the formula for ∫ √(a² – x²) dx?

∫ √(a² – x²) dx = (x/2)√(a² – x²) + (a²/2) sin⁻¹(x/a) + C

State the integration‐by‐parts rule.

∫ f(x) g'(x) dx = f(x) g(x) – ∫ f'(x) g(x) dx

What is the order that helps choose parts for ∫ f g dx?

Its ILATE ( Inverse trigo, logarithmic, algebraic, trigo, exponential

When integrating (sinᵐ x)(cosⁿ x), which substitution is used if m is odd?

Let sin x = t (keep one sine factor for dt) if m is odd.

In ∫ (sinᵐ x)(cosⁿ x) dx, what do you do when n is odd?

Let cos x = t (keep one cosine factor for dt) if n is odd.

How are even–even powers of sin and cos usually handled?

Use half‐angle identities (e.g., sin²x = (1 – cos2x)/2).

Which substitution simplifies ∫ secᵐ x tanⁿ x dx with n odd?

Let sec x = t when tan exponent is odd.

Give the shortcut: ∫ [x f'(x) + f(x)] dx.

∫ [x f'(x) + f(x)] dx = x f(x) + C

Provide the exponential shortcut: ∫ eˣ[f(x) + f'(x)] dx.

∫ eˣ[f(x) + f'(x)] dx = eˣ f(x) + C

What is the first step if deg(Numerator) ≥ deg(Denominator) in a rational integral?

Perform long division to reduce the degree.

When can partial fractions be applied directly to ∫ P(x)/Q(x) dx?

When deg(Q) > deg(P) after any needed division.

How is a linear factor repeated twice handled in partial fractions?

Include terms A/(x – r) + B/(x – r)² for the repeated root r.

Which trig substitution is standard for √(a² – x²)?

Set x = a sin θ (or a cos θ).

What substitution tackles √(a² + x²)?

Take x = a tan θ.

How do you treat √(x² – a²) in an integral?

Use x = a sec θ (or a cosec θ).

State the result of ∫ 1/(1 + x²) dx.

∫ 1/(1 + x²) dx = tan⁻¹ x + C

Give the final general pattern for ∫ (P x + Q)/(ax² + bx + c) dx after splitting.

Integral = k₁ x + k₂ ln|ax² + bx + c| + C, where k₁, k₂ are constants determined from P and Q.

How can ∫ (sin x – cos x) / (sin x + cos x) dx be simplified?

Set t = sin x + cos x so that dt = (cos x – sin x) dx.