MATH 1220, Spring 2005
(2) ( § 7.1) Calculate:
Z x x
√ x + 1
(1) ( § 7.1) Calculate: D x ln
3 x 2 ln x 3
+ 1 dx .
.
Z
1 x + 1
(3) ( § 7.1) Calculate: x 2 + 2 x + 1 dx .
(4) ( § 7.2) For f ( x ) = x − 1 find
(5) ( § 7.3) Calculate: D x e x
2
+ln x f
.
− 1 ( x ) and verify that f − 1 ( f ( x )) = x and f ( f − 1 ( x )) = x .
Z
(6) ( § 7.3) Calculate: xe x
2
+1 dx .
(7) ( § 7.4) Calculate: D x log
2
( x 3
(8) ( § 7.4) Calculate: D x
(2 3 x
+ 1).
+ (3 x ) 2 ).
(9) ( § 7.4) Calculate: D x
( x 2 + 1) ln x .
(10) ( § 7.5) A radioactive substance has a half-life of 800 years. If there were 10 grams initially, how much would be after 150 years ? (Deduce the equation of the exponential decay).
(11) ( § 7.6) Solve the differential equation: xy ′ + (1 + x ) y = e − x ; y = 0 when x = 1.
(12) ( § 7.6) A tank initially contains 100 gallons of brine with 25 pounds of salt solution. Brine containing 1 pound of salt per gallon is entering the tank at the rate of 2 gallons per minute and is flowing out at the same rate.
If the mixture in the tank is kept uniform by constant stirring, find the amount of salt in the tank at any time t .
(13) ( § 7.7) Calculate: D x x tan −
1 x 2
2
.
(14) ( § 7.7) Show the identity: tan(sin
(15) ( § 7.8) Calculate:
Z z
√ z dz .
− 1 x ) = √ x
1 − x 2
.
(16) (
(17) (
§
§
8.1) Calculate:
8.1) Calculate:
Z
Z e x
√
1 − e
1
2 x
2 x
√
4 x 2 dx
− 1
.
dx .
Z
(18) ( § 8.2) Calculate: sin 3 (2 x ) p cos(2 x ) dx .
(19) ( § 8.2) Calculate:
Z sin x cos(2 x ) dx .
(20) ( § 8.3) Calculate:
Z
1
0
√ t
3 t + 1 dt .
(21) ( § 8.3) Which trigonometric substitution is used in order to compute the integral: the integral with the new variable.
(22) ( § 8.4) Calculate:
Z ln(2 x 4 ) x 2 dx .
Z
(23) ( § 8.4) Calculate: x
2 e x dx .
Z
√ x 2
2 x + 1
+ 4 x + 5 dx ? Write
1