Ch 11 Integrals Notes

# Ch 11 Integrals Notes - 1.a CHAPTER 11 Indefinite...

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1 CHAPTER 11 Indefinite Integral (requires a constant of integration, C) 2 (2x 7)dx x 7x C Definite Integral 6 6 2 2 2 x x dx 18 2 16 2 A definite integral can represent the area under the curve if it is all above or below the x axis or the difference in the areas if the x axis is intercepted. It does not require a ‘C’ as the ‘C’ would cancel anyway. Fundamental Theorem of Calculus b / a [f (x)] dx F(b) F(a) where F (x) f (x) and f(x) is continuous for [a, b] Examples: 1. 5 dx 5x C 2. 2 3 1 x dx x C 3 3. n n 1 a ax dx x C, n 1 n 1 4. 1 1 x dx dx ln x C x 5. sin x dx cosx C 6. 1 1 cos x dx 4sin x C 4 4 7. 1 sec5x tan5x dx sec5x C 5 8. 4x 4x 1 e dx e C 4 9. x x 1 a dx a C lna 10. k f(x) dx k f(x) dx 5 5 0 18 -2 1.a

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2 11. f(x) g(x) dx f(x) dx g(x) dx 12. 2 x x dx x 1 1 2 2 2 x x x dx 2x C 2 13. 2 0 2cos3 d 2 0 2sin3 2 2 0 3 3 3 14. 2 2 4 5 2 1 2 2 4 9 3x dx 9x 12x dx x 6x 4x C x 5 x 15 2 2 x 7x 10 (x 2)(x 5) (x 5) 1 5 dx dx dx x x C 2x 4 2(x 2) 2 4 2 16. 3 2 2 2 1 tan x tan x dx tan x[1 tan x] dx (tan x)[sec x]dx tan x C 2 Substitution Method Recall the chain rule: Given y = F[g(x)], then y / = F / [g(x)][g / (x)] therefore, / / F (g(x))g (x)dx F(g(x)) C Let u = g(x), then / F(g(x)) C F(u) C F (u)du therefore, / / / F (g(x))g (x)dx F (u)du if u = g(x) The substitution method gives us a rational way of dealing with the term generated when we apply the chain rule to the original function. eg 1. 3 2 3 4x x 3 dx 4 3 4 3 2 3 3 3 4 4 4 u 1 x 3 [4x dx] u du u du C x 3 C 3 3 3 4 3 let u = x 3 3 du dx = 3x 2 du = 3x 2 dx 2 4 du 4x dx 3 1.b