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Solution 11804: The Anti-Derivative of Sin(x)*Cos(x) Returns an Expression in Cos(x) Instead of Sin(x) on the TI-89 Family, TI-92 Plus and Voyage™ 200.

Why does the anti-derivative of sin(x)*cos(x) returns -(cos(x))^2/2 instead of 1/2sin^2(x) on the TI-89 family, TI-92 Plus and Voyage 200?

The TI-89 family, TI-92 Plus and Voyage 200 calculators use a rule-based system when computing the indefinite integral of sin(x)*cos(x) or cos(x)*sin(x). Listed below are the steps taken when computing problems such as these:

• The integrand is simplified. Internally, the integrand is stored as cos(x)*sin(x). TI could have chosen to store the integrand as sin(x)*cos(x), but either ordering is equally acceptable.

• Then the algorithm attempts to verify whether the integrand is of the form const*elementary_function(u(x))*u'(x).

Here, the first factor is cos(x), so u(x) is cos(x), elementary_function(z) = z, and u'(x) = -sin(x).

Consequently, the answer reported is -(cos(x))^2/2.

Please see the TI-89 family, TI-92 family and Voyage 200 guidebooks for additional information.