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Mathematical Expression Editor
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Various exercises relating to the integration of trigonometric functions.
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.
(Your answer should not include any constant term.)
(Your answer should not include any constant terms.)
(Add a constant to your answer if needed so that it equals at .)
(Add a constant to your answer if needed so that it equals at .)
(Your answer should not include any constant terms.)
You can either use an integration by parts technique or you can use a trigonometric
identity to simplfy the expression (for constants and ) as a sum of simpler
things.
(Your answer should not include any constant terms.)
(Your answer should not include any constant terms and should equal at .)
Use
power reduction formulas.
To fully evaluate the integral from Example trig:reduce_example, it helps to identify the pattern.
Suppose that the power is replaced by an unknown positive constant . Carry out the
calculation again with the unspecified exponent: We conclude Using this formula
several times in a row gives the result
Sample Quiz Questions
Compute the value of the integral (Hints won’t be revealed until after you choose a
response.)
To simplify the calculation, begin with a substitution which replaces with . The
question reduces to computing This integral is compatible with the substitution .
By the substitution formula, this means , and one must also replace by .
Furthermore, by virtue of the special angle formulas and , the problem is reduced to
computing the integral
Carrying out this calculation in the usual way gives a final
answer of .
Compute the value of the integral (Hints won’t be revealed until after you choose a
response.)
Since the power of secant is odd and the power of tangent is even, try rewriting the
integral in terms of sine and cosine. This gives This integral is compatible with the
substitution .
By the substitution formula, this means . Furthermore, by virtue of
the special angle formulas and , the problem is reduced to computing the
integral
Carrying out this calculation in the usual way gives a final answer of
.
Compute the value of the integral (Hints won’t be revealed until after you choose a
response.)
This integral is compatible with the substitution . By the substitution formula,
this means , and one must also replace by . Furthermore, by virtue of the
special angle formulas and , the problem is reduced to computing the integral
Carrying out this calculation in the usual way gives a final answer of
.
Sample Exam Questions
Compute the integral below.
Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)
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Start typing the name of a mathematical function to automatically insert it.
(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)