Chapter 4: Integration
Section 4.3: Fundamental Theorem of Calculus and the Indefinite Integral
Evaluate the indefinite integral ∫5 sec2x−3 sinx ⅆx.
Solution via Maple
Control-drag the indefinite integral.
Context Panel: Evaluate and Display Inline
∫5 sec2x−3 sinx ⅆx = 5⁢sin⁡xcos⁡x+3⁢cos⁡x
Solution from first principles
∫5 sec2x−3 sinx ⅆx
=5∫sec2x ⅆx−3∫sinx ⅆx
=5 tanx+3 cosx
Integration is a linear operation, just like differentiation. Therefore, it "goes past" constants, and plus and minus signs, resulting in the right-hand side of the first equation. The right-hand side of the second equation results from finding antiderivatives of sec2x and sinx by consulting, for example, Table 3.10.1
in Section 3.10. The rest is arithmetic.
Note that Maple does not tack on the additive constant. In other words, Maple returns an antiderivative, not the most general antiderivative. This is consistent with the space-saving requirements of printed tables of integrals, but not necessarily with practice in the typical calculus text.
Note, finally, that Maple's internal routines for finding the antiderivative of sec2x causes it to express tanx as sinx/cosx.
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