University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2024
Views:

364

Pages:

14
Author:

Campbell Wagner

**Subject: Integration** **Part B** ∫adx = ax + c ∫xⁿdx = xⁿ⁺¹ / (n+1) + c ∫[f(x) ± g(x)]dx = ∫f(x)dx ± ∫g(x)dx Examples: ∫(3x² - x³ + 5)dx = x³ - x⁴/4 + 5x + c ∫2x(3x-1)²dx = ∫2x(9x² - 6x + 1)dx = ∫(18x³ - 12x² + 2x)dx = 18x⁴/4 - 12x³/3 + 2x²/2 + c ∫√(x²+4)dx = ∫(x²+4)^(1/2)dx = (x²+4)^(3/2)/3 + 4x^(1/2) + c = 10x^(7/2) + 5x^(1/2) + c **Remarks:** 1. ∫sin(ax+b)dx = -1/a cos(ax+b) + c 2. ∫cos(ax+b)dx = 1/a sin(ax+b) + c 3. ∫sec²(ax+b)dx = 1/a tan(ax+b) + c 4. ∫csc²(ax+b)dx = -1/a cot(ax+b) + c 5. ∫sec(ax+b)tan(ax+b)dx = 1/a sec(ax+b) + c 6. ∫csc(ax+b)cot(ax+b)dx = -1/a csc(ax+b) + c Examples: ∫cos(2x-3)dx = 1/2 sin(2x-3) + c ∫sin2x/cosx dx = ∫2sinxcosx/cosx dx = -2cosx + c **Subject:** ∫(1 + sin²θ)cosθdθ = ∫(1 + sin²θ)d(sinθ) = θ + (sin³θ)/3 + c ∫sec²(x)(sec(x) + tan(x))dx = ∫sec²(x)sec(x)dx + ∫sec(x)tan(x)dx = 1/2 tan²(x) + 1/2 sec(x)tan(x) + c ∫sin⁴x dx = ∫sin⁴x * 1 dx = ∫sin⁴x * (1 - cos²x) / (1 - cos²x) dx = ∫(sin⁴x - sin⁴xcos²x) / (1 - cos²x) dx = ∫(tan⁴xsec²x - tan²xsec²x) dx = 1/4 tan⁴x - 1/2 tan²x + c ∫sin(5x)cot(5x)dx = ∫sin(5x)cos(5x)/sin(5x)dx = ∫cos(5x)dx = 1/5 sin(5x) + c ∫(sinx + cosx)²dx = ∫sin²x + 2sinxcosx + cos²x dx = ∫(1 + sin2x)dx = x + (-1/2)cos2x + c = x - 1/2 cos2x + c ∫(cos²(4x) + tan²(5x))dx = ∫(1/2 + 1/2cos8x) + (sec²(5x) - 1)dx = 1/2x + 1/16sin8x + 1/5tan5x - x + c = -1/2x + 1/16sin8x + 1/5tan5x + c