Mathematics 54 I. A. Find Z 1. Z 2. Z 3. Z 4. Z 5. Z 6. Z 7. Z 8. Z 9. Z 10. Z 11. Z 12. Z 13. Z 14. Z 15. Z 16. Z 17. Z 18. Z 19.
Exercises
the following antiderivatives. x csc2 x dx z cosh−1 z dz cos 4x cos 3x dx cos 7α sin 3α dα x3 √ dx 1 − x2
sin2 (4y + 1) cos2 (4y + 1) dy
20.
Z
z 3 + 2z 2 + 3z − 2 dz (z 2 + 2z + 2)2
tan3 (ln x) sec8 (ln x) dx x Z √ 2m − m2 22. dm m Z 23. e2x ln(ex + 1) dx Z √ 24. ( sin θ + cos θ)2 dθ
21.
Z
tan4 (cosh−1 2x) √ dx 4x2 − 1 Z sin 4θ cos 3θ dθ 26. cos 2θ Z p 27. 16 − e2x ex dx Z y csc3 (tanh y 2 ) dy 28. cosh2 y 2 Z 4x − 3 dx 29. x2 − 2x − 3 q Z tan5 (sinh−1 y) sec(sinh−1 y) p 30. dy y2 + 1 Z csc4 x dx 31. cot2 x Z 45 − 4z 32. dz 2 (z + 2)(3 − z)2 Z p 33. 9 + cos2 x sin x dx Z v 34. ee +v cos ev dv Z x3 − 2x dx 35. (x2 + 1)2 Z 2t dt 36. (2t − 2)(4t + 1) 25.
Z
cos θ sin θ dθ
Z
π 2
5.
√ x2 − 4 dx x
Z
0
6.
w dw w2 + 4w + 13
Z
0
7. 8.
Z
+∞
sin2 (πx) cos2 (πx) dx
3x2 − 1 √ tan−1 x dx 2x x y 3 + 2y 2 + 2 dy y (y 2 + 1)2 cos3 x √ dx sin x sin3 θ sin2 3θ + √ dθ cos θ ln(x2 + 1) dx w2 − 6w + 3 dw w2 (w2 + 3) 1 √ dx x2 + 2x − 15
ew sec4 ew tan4 ew dw 1 dx x3 (x − 1)
sinh z ln2 (cosh z) dz 1 dx (16 + x2 )2 csc6 (log5 t) cot2 (log5 t) dt t sin(ln x) dx
B. Evaluate the following integrals. π 2
1.
Z
4
2.
Z Z
1
Z
1
0
2
3.
0
4.
0
3
√ 5
ex sin x dx
0
coth t dt
−1
−∞
1
1 dx x2 + 16 1 √ dv v 4v 2 − 1
∞
1 + x2 dx 2 −∞ 1 − x Z 1 1 10. dx −x − ex 0 e Z 4 5 11. (s2 − 6s + 10)− 2 ds 9.
Z
Z
2
12.
+∞
13.
Z
2
14.
Z
0
1 dx (2x − 1)2/3 2
r er dr
0
0
−∞
3x + 2 dx (x − 2)(x2 + 4)
II. 1. Let f be a function continuous on [1, 4] whose graph is tangent to the x-axis at x = 1. If f (4) = 3 Z 4 and f ′ (4) = 2, determine the value of xf ′′ (x)dx. 1 Z π 2. Show that cos(mx) cos(nx) dx = 0 for positive integers m and n with m 6= n. −π
1 x−a 1 + C, |x| > a > 0 using dx = ln 3. Show that x2 − a2 2a x+a a. trigonometric substitution; b. partial fractions. Z +∞ 1 4. Show that p dx is convergent if p > 1 and is divergent if p ≤ 1. x(ln x) 2 Z
III. A. Solve the following differential equations. d2 y 1 = √ dx2 x x2 − 1 dy y cos x 6. , y(0) = 0 = dx 1 + y2 7. xy ′ + y = y 2 , y(1) = −1
1. 3dy + yx4 dx = 2y 3 dy dy xex 2. = p dx y 1 + y2
5.
ey sin2 x dy = dx y sec x x d2 y = 4. dx2 (4x2 + 1)2
3.
B. Solve the following problems completely. 1. Find the equation of the curve that passes through the point Q(0, 1) such that the slope of the tangent line to the curve at P (x, y) is xy. d2 y 2. The points Q1 (1, 3) and Q2 (0, 2) are on a curve, and at any point P (x, y) on the curve, = dx2 2 − 4x. Find the equation of the curve. 3. Find the function f such that f ′ (x) = f (x)(1 − f (x)) and f (0) = 21 . 4. The transport of a substance across a capillary wall in lung physiology has been modeled by the differential equation dh h R , =− dt V k+h where h is the hormone concentration in the bloodstream, t is time, R is the maximum transport rate, V is the volume of the capillary, and k is a positive constant that measures the affinity between the hormones and the enzymes that assist the process. Solve this differential equation to find a relationship between h and t. 5. Find the orthogonal trajectories of the family of curves. a. x2 + 2y 2 = K 6. A a. b. c. d.
b. y 2 = Kx3
c. y =
x 1 + Kx
bacteria culture starts with 500 bacteria. After 3 hours, there are 8000 bacteria. Find: the initial population; the formula for the population after t hours; the number of bacteria after 5 hours; the rate of growth after 5 hours;
e. the time when the population reaches 200000. 7. The half-life of cesium-137 is 30 years. Suppose we have a 100-mg sample. a. Find the mass that remains after t years. b. How much of the sample remains after 100 years? c. After how long will only 1 mg remain? 8. After 3 days, a sample of Radon-222 decays to 58% of its original amount. Find: a. the half-life of Radon-222; b. the time it would take for the sample to decay to 10% of its original amount.