tailieunhanh - Handbook of mathematics for engineers and scienteists part 46

Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 46', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | . Indefinite Integral 283 are evaluated using the formulas sin a cos 3 2 sin a 3 sin a - 3 cos a cos 3 2 cos a 3 cos a - 3 sin a sin 3 2 cos a - 3 - cos a 3 . 4. Integrals of the form J sin x cos x dx where m and n are integers are evaluated as follows a if m is odd one uses the change of variable cos x z with sin x dx -dz b if n is odd one uses the change of variable sinx z with cos x dx dz c if m and n are both even nonnegative integers one should use the degree reduction formulas sin2x 2 1 -cos2x cos2 x 2 1 cos2x sinxcosx 2 sin2x. Example 3. Evaluate the integral J sin5 x dx. This integral corresponds to odd m m 5. With simple rearrangement and the change of variable cos x z we have y y x - xdx -ya .ad. .y0 - dz z3 - z5 - z C cos3 x - cos5 x - cos x C. Remark. In general the integrals J sinp x cosq x dx are reduced to the integral of a differential binomial by the substitution y sin x. . Integration of Polynomials Multiplied by Elementary Functions Throughout this section Pn x designates a polynomial of degree n. . Integration of the product of a polynomial by exponential functions. General formulas Pn x eax dx eax Pn x a pn x . t nn e x i i c ypn x cosh ax sinh ax Pn x si h ax dx cosh ax Pn x P x a Pn x P4 x O - cosh ax - sinh ax P x P3l x r p x f P x a c C. These formulas are obtained by repeated integration by parts see formula 4 from Paragraph with f x Pn x for g n 1 x eax g n 1 x cosh ax and g n 1 x sinh ux respectively. In the special case Pn x xn the first formula gives xneax dx eax V -1 n n xk C. J an 1-k k J k 0 284 Integrals . Integration of the product of a polynomial by a trigonometric function. 1 . General formulas fp. x cos ax sn lPn x sin ax sin ax Pn x a Pn X a2 Pn X a3 Pn x a4 cos ax - cos ax - n x Pn x a2 a4 - P. X P x C . a3 C a These formulas are obtained by repeated integration by parts see formula 4 from Paragraph with f x Pn x for g n 1 x cos ax andg n 1 x sin ax respectively. 2 . To evaluate integrals