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Mathematical Expression Editor

Various exercises relating to partial fractions and integration.

Compute the indefinite integrals below. Since there are many possible answers (which
differ by constant values), use the given instructions if needed to choose which
possible answer to use. Do not forget absolute value signs inside logarithms when
they are needed.

(Do not include any constant terms in your antiderivative.)

(Do not include any constant terms in your antiderivative.)

(Do not include any constant terms in your antiderivative.)

(Do not include any constant terms in your answer.)

(Do not include any constant terms in your answer.)

Don’t forget polynomial long
division; it is needed in this case because the degree of the numerator is at least as
large as the degree of the denominator.

(Do not include any constant terms in your answer.)

Since the derivative of the
denominator is , we should rewrite the numerator of the big fraction to have ’s
if possible: For expressions like we should do a substitution. For terms
like we should first complete the square: and then make the substitution
.

(Do not include any constant terms in your answer.)

Sample Quiz Questions

Compute the integral (Hints won’t be revealed until after you choose a response.)

First factor the denominator of the integrand: . Since the roots are distinct, it is
possible to use the Heaviside cover-up method.

The partial fractions expansion will
take the form where the coefficient can be computed by cancelling the factor of in
the denominator and evaluating the result at , i.e., Similarly, which gives that

Therefore

Sample Exam Questions

Compute the volume of the solid of revolution obtained by revolving around
the -axis the region below the graph above , and between and . (Hints
won’t be revealed until after you choose a response.)

Choosing as the variable of integration, slices will be parallel to the -axis, indicating
that the shell method should be used. The radius of a shell is (because the axis lies
to the left of the region) and the height will be , so

The integral can be computed
by partial fractions; the expansion has the form The coefficients and can
be found by usual methods (but note that the Heaviside cover up method
will not work in this case), but it is also possible to find them directly by
carefully rewriting the numerator of the fraction in terms of : Therefore

Compute the constants and in the partial fractions expansion indicated
below. To receive full credit, it is not necessary to compute or . (Hints
won’t be revealed until after you choose a response.)

You’ll need to do polynomial long division first. To compute , you can use Heaviside
cover-up.