Review for Midterm 1
Math 153H
1
Integrals
Evaluate the following:
R
1. tan3 (x) dx
R √
3
2. e x dx
R 2
3. ex2x dx
R
1
4. √2x−x
dx
2
R
1
5. x2 +4x+9
dx
R
6. sin(ln(x)) dx
R√
5 + 2x + x2 dx
7.
R1
8. 0 sin−1 (x) dx
R1
9. 0 e3x sin(3x) dx
Re
10. 1 x5 (ln(x))2 dx
R π/4
11. −π/4 tan2015 (x) dx
2
Hyperbolic Trig Functions
Consider the six hyperbolic trig functions
sinh(x),
cosh(x),
tanh(x),
sech(x),
csch(x),
coth(x)
1. Express each of the hyperbolic trig functions in terms of ex and e−x .
2. Write the derivative of each of hyperbolic trig function.
3. Write the derivative of the inverse of each hyperbolic trig function.
4. Simplify cosh(sinh−1 (2x)) (write without using hyperbolic functions).
5. Use the expression for cosh(x) in terms of ex to show that cosh(x) is positive for all
x.
6. Use calculus to show that sinh(x) is increasing, and find where sinh(x) is concave up
and concave down. Use this information to sketch the graph of sinh(x).
7. Sketch the graph of cosh(x).
3
Differential Equations
1. Find all solutions to the differential equation x00 (t) − 81x(t) = 0.
2. Graph the solution x(t) to the equation x00 (t) + 9x(t) = 0 with initial conditions
x(0) = 0, x0 (0) = 4.
3. Find all solutions y = y(x) to the differential equation x2 y 0 + y = 0. You may assume
x > 0 and y 6= 0.
0
4. Find all solutions y = y(x) to the differential equation y 00 = ey +x .
4
Miscellaneous
1. Calculate the area of the region bounded by the graphs of f (x) = 4x and g(x) = x3 .
R tan(x)
d
2
2. Evaluate dx
sin(x) cos(sin(t )) dt.