Math: Calculus 2 — Integration Techniques

What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv - ∫v du.

What does the acronym LIATE stand for, and what is it used for in Calculus?

LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. It is a rule of thumb used to choose the 'u' function in integration by parts.

When evaluating the integral ∫sinⁿ(x)cosᵐ(x) dx, what strategy should you use if the power of sine (n) is odd and positive?

If the power of sine (n) is odd, save one sine factor to serve as du, and convert the remaining even power of sine to cosines using the identity sin²(x) = 1 - cos²(x), then use u-substitution with u = cos(x).

Which trigonometric substitution should be used to evaluate an integral containing the radical √(a² - x²)?

For an integral containing √(a² - x²), use the substitution x = a sin(θ), which simplifies the radical using the identity 1 - sin²(θ) = cos²(θ).

Which trigonometric substitution is appropriate for an integral involving the expression √(x² + a²)?

For an integral involving √(x² + a²), use the substitution x = a tan(θ), which leverages the identity 1 + tan²(θ) = sec²(θ) to simplify the radical.

In partial fraction decomposition, what is the correct form of the numerator for an irreducible quadratic factor (ax² + bx + c) in the denominator?

For an irreducible quadratic factor (ax² + bx + c) in the denominator, the numerator must be a linear expression of the form Ax + B.

What defines a Type 1 improper integral?

A Type 1 improper integral is an integral where at least one of the limits of integration is infinite (e.g., ∫ from a to ∞, ∫ from -∞ to b, or ∫ from -∞ to ∞).

Under what condition does the p-series Σ (1/nᵖ) converge?

The p-series Σ (1/nᵖ) converges if p > 1 and diverges if p ≤ 1.

What is the condition for a geometric series Σ arⁿ to converge, and what is its sum if it does?

A geometric series converges if the absolute value of the common ratio |r| < 1. If it converges, its sum is a / (1 - r), where 'a' is the first term.

What is the conclusion of the Ratio Test if the limit L = lim (n→∞) |a_{n+1} / a_n| equals 1?

If the limit L = 1 in the Ratio Test, the test is inconclusive, and another convergence test must be used to determine if the series converges or diverges.