Elements of the Differential And Integral Calculus With Applications

William S William Shaffer Hall

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Ans. Sin x — sin^ a; + | sin^ ^ ~ T sin^ a;. . J cot* a; da?. Ans. — -^cot^aj^ cota; + a;.
11. Rcos*a;da;= r(|4-icos2a;)2da; = Ja; + isin2x4-i rcos'(2a;)c?(2a;) = Ja; + Jsin2a; + \ [x + Jsin4a;] = fa; + ^sin2a; + ■ji2^sin4a;.
Art. 34. Definite Integrals.
It was shown in Art. 32, that an arbitrary constant must be added after each integration. Before the value of this constant is determined, the integral is said to be indefinite.
If, from the data of the given problem, the value of the integral ...is known for some particular value of the variable, the constant can be determined by substituting this value of the variable in the indefinite integral.
For example, let ds — = gt -\-v', m which g and v' are constants.
10 50 DIFFERENTIAL AND INTEGRAL CALCULUS.
By integrating and adding the constant C, ^^ Now if S = S' when t = 0, C will be equal to S', and. S = igt^-{-v't-\-S'.
If, in any indefinite integral, two different values of the variable be substituted for the variable, and the result given by the second substi- tution be subtracted from the first result, the constant of integration is eliminated, and the integral is said to be taken between limits.

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