Name _______Answers_____________                        Calculus                        Quiz 4 (Bonus)

`Problem 1.`

``

Int(5*x^(2/3)-2/x^(2/3),x) = 3*x^(5/3)-6*x^(1/3)+C

h(x) = 3*x^(5/3)-6*x^(1/3)

`Check:  `*`h'`(x) = D(3*x^(5/3)-6*x^(1/3))

`h'`(x) = 5*x^(2/3)-2/x^(2/3)

`Problem 2. Find `*f(x)

`f ''`(x) = 2*exp(x)-3*cos(x), f(0) = 1, `f '`(0) = 2

Int(2*exp(x)-3*cos(x),x) = h(x)+C

Int(2*exp(x)-3*cos(x),x) = 2*exp(x)-3*sin(x)+C

`f '`(x) = 2*exp(x)-3*sin(x)+C

`f '`(0) = 2+C

2 = 2+C

C = 0

`f '`(x) = 2*exp(x)-3*sin(x)

f(x) = 2*exp(x)+3*cos(x)+D

f(0) = 5+D

1 = 5+D

D = -4

f(x) = 2*exp(x)+3*cos(x)-4

`Problem 3.`

``

Int((4+x^2)/x^3,x) = h(x)+C

``

Int((4+x^2)/x^3,x) = -2/x^2+ln(x)+C

`Check:  `*`h'`(x) = D(-2/x^2+ln(x))

`h'`(x) = 4/x^3+1/x

``

`Evaluate Definite Integral:  `*h(x) = -2/x^2+ln(x)

Int((4+x^2)/x^3,x = 1 .. 3) = h(3)-h(1)

`` = 16/9+ln(3)

`` = 2.876390067

`Problem 4.`

``

Int(3*x^2*exp(x^3),x) = h(x)+C

h(x) = ``

``

Int(3*x^2*exp(x^3),x) = exp(x^3)+C

h(x) = exp(x^3)

`Check:  `*`h'`(x) = D(exp(x^3))

`h'`(x) = 3*x^2*exp(x^3)

`Problem 5.`

``

Int(exp(cos(x))*sin(x),x) = h(x)

Int(exp(cos(x))*sin(x),x) = -exp(cos(x))

h(x) = -exp(cos(x))

`Check:  `*`h'`(x) = D(-exp(cos(x)))

`h'`(x) = exp(cos(x))*sin(x)

``

`Evaluate Definite Integral:  `*h(x) = -exp(cos(x))

Int(exp(cos(x))*sin(x),x = 0 .. 1/2*Pi) = h(1/2*Pi)-h(0)

`` = exp(1)-1

`` = 1.718281828

`Problem 6.`

``

Int((4+3*x)^(1/2),x) = h(x)+C

Int((4+3*x)^(1/2),x) = 2/9*(4+3*x)^(3/2)+C

h(x) = 2/9*(4+3*x)^(3/2)

`Check:  `*`h'`(x) = D(2/9*(4+3*x)^(3/2))

`h'`(x) = (4+3*x)^(1/2)

``

`Evaluate Definite Integral:  `*h(x) = 2/9*(4+3*x)^(3/2)

Int((4+3*x)^(1/2),x = 0 .. 8) = h(8)-h(0)

`` = -16/9+112/9*7^(1/2)

`` = 31.14712741