Calculus 2 Integration Techniques

Overview

This lecture covers key integration techniques from Calculus 2, including integration by parts, trigonometric integrals, and trigonometric substitution, with step-by-step examples for each method.

Integration by Parts

• Use integration by parts for the integral of two multiplied functions: ∫u dv = u·v − ∫v du.
• Example: ∫x sin(x) dx. Let u = x, dv = sin(x) dx, so du = dx, v = −cos(x).
• Solution: −x cos(x) + ∫cos(x) dx = −x cos(x) + sin(x) + C.
• Example: ∫ln(x) dx. Let u = ln(x), dv = dx, so du = (1/x) dx, v = x.
• Solution: x ln(x) − ∫x·(1/x) dx = x ln(x) − x + C.

Trigonometric Integrals

• For powers of trig functions, use identities and substitution.
• Example: ∫cos³(x) dx. Write as ∫(cos²(x)·cos(x)) dx, replace cos²(x) with 1 − sin²(x).
• Substitute u = sin(x), du = cos(x) dx, integrate to get sin(x) − (1/3)sin³(x) + C.
• For ∫cos⁵(x) sin⁴(x) dx, split cos⁵(x) as cos(x)·(cos²(x))², substitute using identities and u = sin(x).
• After expansion and integration, result is (1/5)sin⁵(x) − (2/7)sin⁷(x) + (1/9)sin⁹(x) + C.
• For ∫sin²(x) dx, use the half-angle identity: sin²(x) = ½(1 − cos(2x)).
• Integrate to get (1/2)x − (1/4)sin(2x) + C.

Tangent and Secant Integrals

• Example: ∫tan⁶(x) sec⁴(x) dx. Use identity sec²(x) = 1 + tan²(x), and u = tan(x).
• Split sec⁴(x) into sec²(x)·sec²(x), substitute and expand.
• Result: (1/7)tan⁷(x) + (1/9)tan⁹(x) + C.

Trigonometric Substitution

• Use trigonometric substitution for integrals involving √(a² − x²), √(a² + x²), or √(x² − a²).
• For √(a² − x²), let x = a sin(θ).
• Example: ∫√(4 − x²)/x² dx. Substitute x = 2 sin(θ), dx = 2 cos(θ) dθ.
• Simplify using trig identities; expression becomes an integral in terms of cot²(θ) and dθ.
• Integrate to get −cot(θ) − θ + C; back-substitute using a reference triangle.
• Final answer: −√(4 − x²)/x − arcsin(x/2) + C.

Key Terms & Definitions

• Integration by Parts — Method for integrating the product of two functions: ∫u dv = u·v − ∫v du.
• Trigonometric Integral — Integral involving powers/products of trig functions, often solved with identities or substitution.
• Half-Angle Identity — sin²(x) = ½(1 − cos(2x)).
• Trigonometric Substitution — Replacing variables with trig functions to simplify integrals containing radicals.

Action Items / Next Steps

• Review and practice similar integration by parts and trigonometric integration problems.
• Study and memorize key trig identities and substitution techniques.
• Check assigned calculus video playlist for additional practice and examples.