Unit 2

Trigonometric Integrals

100

To evaluate the integral of cos(x)ln(sinx)dx using integration by parts, a good choice for u and dv would be...

What is u=ln(sinx), dv=cosxdx

200

Evaluate the integral of dx/(4+x^2)^2 using trigonometric substitution.

What is 1/16 arctan(x/2) + 1/8 (x/(4+x^2)) + C

300

Evaluate the integral of (x^2 - 25)^(1/2) / x dx

What is (x^2 - 25)^(1/2) - 5 arcsec(x/5) + C

Partial Fraction Decomposition

100

Write the appropriate partial fraction decomposition general form for the rational expression (3x-1) / x^2(x^2 + 3)^2

What is A/x + B/x^2 + (Cx+D)/(x^2 + 3) + (Ex+F)/(x^2 + 3)^2

200

Solve the integral of (-2x^2 + 6x + 5) / (x+4)(x^2+1)dx.

What is -3ln(x+4) + 1/2 ln(x^2+1) + 2arctanx + C

300

Evaluate the integral of (3x^2 +x) / (2x-1)(x^2+1) dx.

What is 1/2 ln|2x-1| + 1/2 ln|x^2 + 1| + arctan|x| + C

Limits

100

* Describe how to solve a limit that involves an x to the power of a function.

What is lim e^ (ln(function)) then use properties of ln to solve.

200

Evaluate the limit as x approaches infinity of x^((ln2)/(1+lnx))

What is 2

300

Evaluate the limit as x approaches 1 from the right of x^(1/(x-1))

What is e

Improper Integrals

100

Evaluate the integral from negative infinity to 0 of e^(-3x)dx.

What is the integral diverges.

200

Set up and evaluate (if possible) the integral that gives the volume of the solid formed when the region bounded by the graph of f(x)=1/(x-1)^(1/4), y=0, x=1, and x=2 is revolved around the x-axis. To receive full credit for this problem, the integral must be correctly set up. Hint: Use the disk method.

What is 2pi.

300

Set up and evaluate the integral that gives the area of the region bounded by the graph of f(x)=lnx/x and the x-axis on the interval [1, infinity).

What is the limit does not exist, therefore the improper integral diverges. The area of the region is infinite.