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finalreviewf11

# finalreviewf11 - MAC 2311 Fall 2011 Some Review...

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MAC 2311, Fall 2011: Some Review Problems (Lectures 33 - 35) Final covers lectures 1 - 35 1. Evaluate each integral: a) Z csc ± csc ± ± sin ± (Hint: multiply each term in the fraction by sin ± .) b) Z 0 ± 1 = 2 3 ± 4 p 1 ± x 2 dx c) Z 1 = p 3 0 x 2 ± 1 x 4 ± 1 dx (Hint: simplify the integrand.) 2. Evaluate each integral: a) 3 x x 2 ± 4 dx b) Z 3 x + 3 p x 2 + 2 x + 5 dx c) Z ±= 6 0 sec(2 x )[3 tan(2 x ) + sec(2 x )] dx d) Z e 2 1 (1 + 2 ln x ) 2 x dx e) Z sin(2 x ) 1 + sin 2 x dx f) Challenge! Z cos x 1 + sin 2 x dx g) Z ln2 0 e 4 x ± e x e 2 x dx h) Z cot xdx i) Z e 1 2 x ± 1 (2 x ± 1) 2 dx j) Z 0 ± 2 x p 2 x + 4 dx k) Z e 2 x e x ± 1 dx l) Z ±= 2 0 sin(sin x ) cos xdx * *Be sure to check your integral by di±erentiation, and express your answer as a decimal. 3. Find the maximum and minimum values of f ( x ) = 3 p x 2 + 2 x on [ ± 2 ; 2] and use them to ²nd upper and lower bounds for the de²nite integral Z 2 ± 2 3 p x 2 + 2 xdx . 4. Evaluate Z ( p x + 1) 2 p x dx in two di±erent ways. If the slope of the function y = f ( x ) at any point x is given by ( p x + 1) 2 p x

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