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Chapter 5 | Integration
Example 5.29 Integrating an Odd Function
Evaluate the definite integral of the odd function −5sin x over the interval [− π , π ].
Solution The graph is shown in Figure 5.36 . We can see the symmetry about the origin by the positive area above the x -axis over [− π , 0], and the negative area below the x -axis over [0, π ]. We have ∫ − π π −5sin xdx =−5(−cos x ) | − π π =5cos x | − π π = [5cos π ]− ⎡ ⎣ 5cos(− π ) ⎤ ⎦
=−5−(−5) =0.
Figure 5.36 The graph shows areas between a curve and the x -axis for an odd function.
−2 2
5.24
Integrate the function ∫
x 4 dx .
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