1597, Math Honors Calculus II Test 2 Practice Problems answers 1. a) 1/2 b) e1 2. 10. Each integration by parts decreases the power of x in the integrand by one. After 10 integrations by parts, the remaining integral can be evaluated directly. 3.
3 2 x sin(2x) 3 + 4 cos(2x) + C
4. 1/4. (Let u = cos(x) or u = sin(x). 5. cos(3x) 6 6.
cos(7x) 14
+C
8 tan3 () sec()d
Bx+C x2 +9
7. a) A + x
b) A + x
B x2
+
C x3
+
D x2
8. 2 ln |x| ln |x + 1| + 1 ln |x 1 | + C 2 2 9. 2x ln |x + 2 | + C 3 10.
2u2 du. u2 4 4x2 3 3x
Easier since no square root: divide out and use partial fractions.
11. 12. 2
+C
13. True. {an } must not only converge, but must converge to zero. Lots 14. of correct answers. For example: 1,-1,1,-1,... or an =
(1)n n .
15. 0. Lots of correct explanations. For example, the degree of the denominator is larger than the degree of the numerator, so the denominator grows faster than the numerator. Or, better, divide all terms in numerator and denominator by n2 ; then the numerator goes to zero and the denominator goes to one. 16. (a) converges to 8 ...