Integration Techniques

Example 1 — U-Substitution: ∫ x / (x² + 4) dx

Example 2 — Integration by Parts: ∫ x² sin x dx

Example 3 — Trig Integral: ∫ sin⁵x dx

Example 4 — Trig Substitution: ∫ √(4 − x²) dx

Example 5 — Partial Fractions: ∫ 1 / (x² − 1) dx

How do you know when to use u-substitution?

Use u-substitution when you can identify a composite function inside the integral and its derivative (or a constant multiple of it) also appears in the integrand. Look for a function and its derivative together: ∫f(g(x))g'(x) dx. Set u = g(x) (the inner function), compute du = g'(x) dx, and substitute to simplify. Example: ∫2x·cos(x²) dx — let u = x², du = 2x dx, transforms to ∫cos(u) du = sin(u) + C = sin(x²) + C.

What is the LIATE rule for integration by parts?

LIATE is a mnemonic for choosing u in integration by parts (∫u dv = uv − ∫v du). Choose u to be the type of function that appears earliest in the acronym: L = Logarithmic (ln x, log x), I = Inverse trig (arctan x, arcsin x), A = Algebraic (polynomials like x², 3x + 1), T = Trigonometric (sin x, cos x), E = Exponential (eˣ, 2ˣ). Example: ∫x·eˣ dx — x is Algebraic and eˣ is Exponential; A comes before E, so u = x and dv = eˣ dx. Result: xeˣ − eˣ + C.

When do you use the tabular method for integration by parts?

Use the tabular method (also called the DI method or repeated integration by parts) when you must apply integration by parts multiple times and u is a polynomial. Create two columns: D (differentiate u repeatedly until reaching 0) and I (integrate dv repeatedly). Alternate the signs +, −, +, −, … and multiply diagonally. Example: ∫x³eˣ dx — D column: x³, 3x², 6x, 6, 0; I column: eˣ, eˣ, eˣ, eˣ, eˣ. Answer: x³eˣ − 3x²eˣ + 6xeˣ − 6eˣ + C = eˣ(x³ − 3x² + 6x − 6) + C.

How do you integrate powers of sine and cosine?

For ∫sinᵐx cosⁿx dx: If m is odd, save one sin x, convert the rest using sin²x = 1 − cos²x, then substitute u = cos x. If n is odd, save one cos x, convert the rest using cos²x = 1 − sin²x, then substitute u = sin x. If both m and n are even, use half-angle identities: sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2. Example: ∫sin³x dx = ∫sin²x · sin x dx = ∫(1 − cos²x) sin x dx; let u = cos x, du = −sin x dx → −∫(1 − u²) du = −cos x + cos³x/3 + C.

What are the three trigonometric substitutions and when do you use each?

The three trig substitutions match the three Pythagorean identities. (1) For √(a² − x²): substitute x = a sin θ, dx = a cos θ dθ. The expression becomes a cos θ. Use when the integrand contains a² − x². (2) For √(a² + x²): substitute x = a tan θ, dx = a sec² θ dθ. The expression becomes a sec θ. Use when the integrand contains a² + x². (3) For √(x² − a²): substitute x = a sec θ, dx = a sec θ tan θ dθ. The expression becomes a tan θ. Use when the integrand contains x² − a². Always draw a right triangle to convert back to x at the end.

How does partial fraction decomposition work?

Partial fraction decomposition rewrites a rational function P(x)/Q(x) as a sum of simpler fractions that are easy to integrate. Step 1: If degree of P ≥ degree of Q, perform polynomial long division first. Step 2: Factor the denominator Q(x) completely. Step 3: Write a partial fraction for each factor — A/(ax+b) for linear factors, (Ax+B)/(ax²+bx+c) for irreducible quadratics, and repeated factors require extra terms like A/(ax+b) + B/(ax+b)². Step 4: Clear denominators and solve for the constants by substituting convenient values of x or matching coefficients. Step 5: Integrate each partial fraction — linear factors give logarithms, irreducible quadratics give arctangents.

What is an improper integral and how do you determine if it converges?

An improper integral has either an infinite limit of integration (Type I) or an integrand with a vertical asymptote in the interval (Type II). To evaluate, replace the problematic part with a limit. Type I example: ∫₁^∞ 1/x² dx = lim_{b→∞} ∫₁^b x⁻² dx = lim_{b→∞} [−1/x]₁^b = lim_{b→∞} (−1/b + 1) = 1. This converges. Type II example: ∫₀^1 1/√x dx = lim_{a→0⁺} ∫ₐ^1 x^{−1/2} dx = lim_{a→0⁺} [2√x]ₐ^1 = 2. This converges. If the limit does not exist or is ±∞, the integral diverges. The p-test: ∫₁^∞ 1/xᵖ dx converges if p > 1 and diverges if p ≤ 1.

How do you choose the right integration technique for a given integral?

Follow this decision process: (1) Is it a basic form? Check the antiderivative table first — many integrals are done immediately. (2) Can you simplify? Expand products, split fractions, use trig identities before trying a technique. (3) Is there a composite function with its derivative? Use u-substitution. (4) Is it a product of two different function types? Use integration by parts — apply LIATE to choose u. (5) Does it involve sin^m · cos^n, tan^m · sec^n, etc.? Use trig integral strategies. (6) Does it contain √(a²−x²), √(a²+x²), or √(x²−a²)? Use trig substitution. (7) Is it a rational function (polynomial over polynomial)? Use partial fractions. (8) Does it have an infinite bound or vertical asymptote? It is an improper integral — convert to a limit.

What is the reduction formula for integration by parts with trig powers?

Reduction formulas reduce the power of a trig function by 2 each time, eventually arriving at a base case. For sine: ∫sinⁿx dx = −(1/n)sinⁿ⁻¹x cos x + ((n−1)/n)∫sinⁿ⁻²x dx. For cosine: ∫cosⁿx dx = (1/n)cosⁿ⁻¹x sin x + ((n−1)/n)∫cosⁿ⁻²x dx. For secant (very useful in Calc 2): ∫secⁿx dx = (1/(n−1))secⁿ⁻²x tan x + ((n−2)/(n−1))∫secⁿ⁻²x dx. Apply the formula repeatedly until reaching ∫sin x dx, ∫cos x dx, ∫sec x dx, or ∫sec²x dx — all of which are standard antiderivatives.