Weekly Tutorial 5 for Math. 69.107A,
Elementary Calculus I

1. Evaluate the definite integral of the function f(x) = e^CubeRoot(x) x^-2/3 over the x-interval ranging from 1 to 8.

a) None of these
b) e² - e
c) 3( e² - e)
d) 3e²

2. Evaluate the indefinite integral of the function cos(2t)/sqrt(1 + sin(2t)).

a) ln|1 + sin(2t)| + C
b) sqrt(1 + sin(2t))
c) sqrt(1 + cos(2t))/2
d) sqrt(1+ sin(2t)) + C

3. Evaluate the indefinite integral of the function ln(2 sqrt(x))/x.

a) ln(2t) + C
b) ln(2sqrt(t)) + C
c) ln(2 sqrt(t))²
d) None of these

4. Use Simpson's Rule with n = 4 in order to approximate the value of the definite integral of the function e^1/x over the interval [1,2] to six decimal places.

a) 2.200651
b) 2.017234
c) 2.032147
d) None of these

5. Evaluate the derivative of the function defined by the integral of e^-t2 over the variable range from sqrt(x) to sin(x).

a) e^-sin2(x) - e^-x + C
b) e^-cos2(x) sin(x) - e^-x/(2 sqrt(x))
c) e^-sin2(x) cos(x) - e^-x/(2 sqrt(x))
d) 2 (e^-sin2(x) - e^-x) sqrt(x)

6.Evaluate the definite integral of the function tan(u) between the limits 0 and Pi/4.

a) ln(2)
b) ln(2) / 2
c) ln(2) + C
d) None of these

1. c ...Let u = x raised to the power of 1/3 ... 2. d ...Let u = 1 + sin(2t) ... 3. c ...Let u = ln(2 sqrt(x)) ... 4. a ...Use Simpson's Rule 5. c ...Use Leibnitz's Rule 6. b ...Let v = cos(u) ...