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Use the Fundamental Theorem of Calculus to determine the answer to the problem; do not use the even/odd properties of integration.

Calculus1
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Problem is: \[\int\limits_{-\frac{ \pi }{ 2 }}^{\frac{ \pi }{ 2 }}(\sin^3x \cos x+\sin x \cos x)dx\]
\[\sin ^{2} x is meant (\sin(x))^{2}\] so you are using quotient rule for sin square. and cos x anti-derivative was sin x

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Other answers:

Then plug pi/2 and - pi/2 in the equation
Then take the number when you plug pi/2 subtract the number when you plug -pi/2 and get the answer
Hope this help
@MoonlitFate find the integral using u substitution (see attachment)
1 Attachment
then evaluate the integral from b to a (see attachment 2) verification of answer @ http://www.wolframalpha.com/input/?i=integral+of+sin^3xcosx%2Bsinxcosx+from+-pi%2F2+to+pi%2F2
1 Attachment

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