lecture34a

# lecture34a - Lecture 34(Chapter 5 Sec 34 Indeﬂnite...

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Lecture 34 (Chapter 5, Sec. 34) Indeﬂnite Integrals; Net Change Theorem Note the connections between antiderivatives and the deﬂnite integral from: Fundamental Theorem of Calculus I If f is continuous, then Z x a f ( t ) dt is Fundamental Theorem II Z b a f ( x ) dx = where Indeﬂnite Integrals Deﬂnite Integrals

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2 We rewrite our antiderivative formulas as integrals: 1. Z cf ( x ) dx = c Z f ( x ) dx 2. Z k dx = 3. Z [ f ( x ) § g ( x )] dx = Z f ( x ) dx § Z g ( x ) dx 4. Z x n dx = 5. Z 1 x dx = 6. Z e x dx = 7. Z a x dx = 8. Z cos x dx =
3 9. Z sin x dx = 10. Z sec 2 x dx = 11. Z csc 2 x dx = 12. Z sec x tan x dx = 13. Z csc x cot xdx = 14. Z 1 p 1 ¡ x 2 dx = 15. Z 1 1 + x 2 dx = NOTE: These apply to intervals only. ex. Z x ¡ 2 dx

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4 Evaluate each integral: ex. Z 4 p x 3 + 1 x dx ex. Z (2 x + 4 p 1 ¡ x 2 ¡ e 2 x ) dx
5 Evaluate each integral: ex. Z 4 0 sin ± cos 2 ± ex. Z tan 2 x dx

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6 ex. Z x 2 x 2 + 1 dx Net Change Theorem The integral of a rate of change of a function is the net change of the function on the interval [ a;b ]: Z b a F 0 ( x ) dx =
7 ex.

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lecture34a - Lecture 34(Chapter 5 Sec 34 Indeﬂnite...

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