Calculus: Analytic Geometry and Calculus, with Vectors

(^506) Exponential and logarithmic functions and hence that (9.312) eu = cos z + i sin z (9.313) e u = cos z - i sin z. Adding a ...

9.3 Hyperbolic functions 507 nometric functions, the first and last are reciprocals, the next to the first and the next to the l ...

508 Exponential and logarithmic functions (9.36) sinh-1 x = log (x + x'- T-1)' d sinh-1 x =^1 dx 1 +x2 1 (9.361) cosh-1 x = log ...

9.3 Hyperbolic functions 509 to discover therole of the parameter t is to let r be the vector running from the origin to a parti ...

510 Exponential and logarithmic functions provided that the positive constant k and the coordinate system are suitably determine ...

9.4 Partial fractions 511 9 Supposing that a and b are constants for which b'- a'-, obtain the formula f ea° cosh ht dt =eat(b s ...

512 Exponential and logarithmic functions of the numerator is less than that of the denominator. Letting f(x) be defined by (9.4 ...

9.4 Partial fractions 513 There is, however, an easier way to find 11, B, C directly from (9.431). Putting x = 1 in (9.431) show ...

514 Exponential and logarithmic functions Remark: We should not be too busy to see how this formula leads to another that also a ...

9.4 Partial fractions 515 2 Supposing that p, q, r are three different constants, determine A, B, C such that 1 1 B + C (x-p)(x- ...

516 Exponential and logarithmic functions Remark: Determination of constants 11, B, C, D for which 1 Bx+B Cx+D (x2+1)(x2+4) x2+1 ...

9.4 Partial fractions 517 11 Partial fraction expansions have their principal applications in electrical engineering and elsewhe ...

518 Exponential and logarithmic functions in which it is supposed that a and b are real or complex numbers for which b tea. For ...

9.5 Integration by parts 519 are functionshaving continuous derivatives over intervals appearing in our work, theformula which c ...

(^520) Exponential and logarithmic functions situations in which matters are simplified by inserting a cl which is different fro ...

9.5 Integration by parts 521 We could feel that integration by parts would not enableus to simplify f log x dx, but we can set u ...

522 Exponential and logarithmic functions 3 Derive the formula fx" cos x dx = x" sin x + X"-1 cos x - n(n - I) fx"-2 cos x dx. U ...

9.5 Integration by parts 523 and show that multiplying the numerators and denominators of the right mem- bers of (2) and (3) by ...

524 Exponential and logarithmic functions and is quite near 1 even when n is as small as 4 or 3. Thus, even when n is quite smal ...

9.5 Integration by parts 525 10 Evaluate the integral in f x3 I J 1+xsdx in two different ways, and make the results agree. Firs ...