cos ax dx 1 a sin ax C 424 x cos ax dx 1 a 2 cos ax x a sin ax C 425 x 2 cos ax

Cos ax dx 1 a sin ax c 424 x cos ax dx 1 a 2 cos ax x

This preview shows page 498 - 502 out of 566 pages.

cos ax dx = 1 a sin ax + C 424. x cos ax dx = 1 a 2 cos ax + x a sin ax + C 425. x 2 cos ax dx = 2 x a 2 cos ax + x 2 a - 2 a 3 sin ax + C 426. x n cos ax dx = 1 a x n sin ax + n a 2 x n - 1 cos ax - n ( n - 1) a 2 x n - 2 cos ax dx 427. cos ax x dx = ln | x | - a 2 x 2 2 · 2! + a 4 x 4 4 · 4! - a 6 x 6 6 · 6! + · · · + ( - 1) n a 2 n x 2 n (2 n ) · (2 n )! + · · · + C 428. cos ax dx x n = - cos ax ( n - 1) x n - 1 - a n - 1 sin ax x n - 1 dx 429. dx cos ax = 1 a ln | sec ax + tan ax | + C 430. x dx cos ax = 1 a 2 a 2 x 2 2 + a 4 x 4 4 · 2! + 5 a 6 x 6 6 · 4! + · · · + E n a 2 n +2 x 2 n +2 (2 n + 2) · (2 n )! + · · · + C 431. dx x cos ax = ln | x | + a 2 x 2 4 + 5 a 4 x 4 96 + · · · + E n a 2 n x 2 n 2 n (2 n )! + · · · + C where E n is the n th Euler number E 1 = 1 , E 2 = 5 , E 3 = 61 , . . . Note scaling and shifting Appendix C
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491 432. dx 1 + cos ax = 1 a tan ax 2 + C 433. dx 1 - cos ax = - 1 a cot ax 2 + C 434. 1 - cos ax dx = - 2 2 cos ax 2 + C 435. 1 + cos ax dx = 2 2 sin ax 2 + C 436. cos 2 ax dx = x 2 + sin 2 ax 4 a + C 437. x cos 2 ax dx = x 2 4 + 1 4 a x sin 2 ax + 1 8 a 2 cos 2 ax + C 438. cos 3 ax dx = sin ax a - sin 3 ax 3 a + C 439. cos 4 ax dx = 3 8 x + 1 4 a sin 2 ax + 1 32 a sin 4 ax + C 440. dx cos 2 ax = 1 a tan ax + C 441. x dx cos 2 ax = x a tan ax + 1 a 2 ln | cos ax | + C 442. dx cos 3 ax = 1 2 a sin ax cos 2 ax + 1 2 a ln | tan π 4 + ax 2 | + C 443. dx 1 - cos ax = - 1 a cot ax 2 + C 444. x dx 1 - cos ax = - x a cot ax 2 + 2 a 2 ln | sin ax 2 | + C 445. dx 1 + cos ax = 1 a tan ax 2 + C 446. x dx 1 + cos ax = x a tan ax 2 + 2 a 2 ln | cos ax 2 | + C 447. dx 1 + cos 2 ax = - 1 2 a tan - 1 ( 2 cot ax ) + C 448. dx 1 - cos 2 ax = - 1 a cot ax + C Appendix C
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492 449. dx (1 - cos ax ) 2 = - 1 2 a cot ax 2 - 1 6 a cot 3 ax 2 + C 450. dx (1 + cos ax ) 2 = 1 2 a tan ax 2 + 1 6 a tan 2 ax 2 + C 451. dx α + β cos ax = 2 a α 2 - β 2 tan - 1 α - β α + β tan ax 2 + C, α 2 > β 2 1 a β 2 - α 2 ln β + α + β - α tan ax 2 β + α - β - α tan ax 2 + C, α 2 < β 2 452. dx α + β cos ax = x α - β α dx β + α cos ax 453. dx ( α + β cos ax ) 2 = α sin ax a ( β 2 - α 2 )( α + β cos ax ) - α β 2 - α 2 dx α + β cos ax , α negationslash = β 454. dx α 2 + β 2 cos 2 ax = 1 α 2 + β 2 tan - 1 α tan ax α 2 + β 2 + C 455. dx α 2 - β 2 cos 2 ax = 1 α 2 - β 2 tan - 1 α tan ax α 2 - β 2 + C, α 2 > β 2 1 2 β 2 - α 2 ln α tan ax - β 2 - α 2 α tan ax + β 2 - α 2 + C, α 2 < β 2 456. dx cos n ax = sec ( n - 2) ax tan ax ( n - 1) a + n - 2 n - 1 sec n - 2 ax dx + C Integrals containing both sine and cosine functions 457. sin ax cos ax dx = 1 2 a sin 2 ax + C 458. dx sin ax cos ax = - 1 a ln | cot ax | + C 459. sin ax cos bx dx = - cos( a - b ) x 2( a - b ) - cos( a + b ) x 2( a + b ) + C, a negationslash = b 460. sin ax sin bx dx = sin( a - b ) x 2( a - b ) - sin( a + b ) x 2( a + b ) + C 461. cos ax cos bx dx = sin( a - b ) x 2( a - b ) + sin( a + b ) x 2( a + b ) + C 462. sin n ax cos ax dx = sin n +1 ax ( n + 1) a + C 463. cos n ax sin ax dx = - cos n +1 ax ( n + 1) a + C Appendix C
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493 464. sin ax dx cos ax = 1 a ln | sec ax | + C 465. cos ax dx sin ax = 1 a ln | sin ax | + C 466. x sin ax dx cos ax = 1 a 2 a 3 x 3 3 + a 5 x 5 5 + 2 a 7 x 7 105 + · · · + 2 2 n (2 2 n - 1) B n a 2 n +1 x 2 n +1 (2 n + 1)!
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