Section 8.1 Basic Integration Rules Fitting Integrands to Basic Rules

332460_0801.qxd 11/2/04 3:04 PM Page 519SECTION 8.1 Basic Integration Rules 519EXAMPLE 2Evaluate10Using Two Basic Rules to Solve a Single Integralx 34 x 2 dx.x + 3 2yy =4 − x 21Solution Begin by writing the integral as the sum of two integrals. Then apply thePower Rule and the Arcsine Rule as follows.111x 304 x dx x2 0 4 x dx 3 1 1114 x22 0 4 x dx 22 12 2x dx 3 2 2 x dx 200−1The area of the region is approximately1.839.Figure 8.11xSee Figure 8.1. 4 x 2 12 3 arcsin x 2 1 0 3 1.8392 2 0TECHNOLOGY Simpson’s Rule can be used to give a good approximation ofthe value of the integral in Example 2 (for the approximation is 1.839).When using numerical integration, however, you should be aware that Simpson’sRule does not always give good approximations when one or both of the limits ofintegration are near a vertical asymptote. For instance, using the FundamentalTheorem of Calculus, you can obtain1.990n 10,x 34 x dx 6.213. 2Applying Simpson’s Rule (with n 10) to this integral produces an approximationof 6.889.EXAMPLE 3 A Substitution Involving a 2 u 2xFind 216 x dx. 6STUDY TIP Rules 18, 19, and 20 of thebasic integration rules on the next pageall have expressions involving the sum ordifference of two squares:a 2 u 2a 2 u 2u 2 a 2With such an expression, consider thesubstitution as in Example 3.u f x,SolutionBecause the radical in the denominator can be written in the forma 2 u 2 4 2 x 3 2you can try the substitution u x 3 . Then du 3x 2 dx, and you havex2Rewrite integral.16 x dx 1 3x32 dx6 16 x 1 duSubstitution: u x33 24 32 u 2 1 Arcsine Rule3 arcsin u 4 C 1 3arcsinx34 C.Rewrite as a function of x.