Интеграл

0.5. Òðèãîíîìåòðèéí èíòåãðàë 100 sin3x cos4xdx èíòåãðàëûã áîä. u = cos x, du = − sin xdx. I = sin3x cos4xdx = sin2x cos4x sin xdx. sin2x = 1 − cos2x. 1 − cos2x cos4x sin xdx I = sin2x cos4x sin xdx = = − 1 − u2 u4du = u6 − u4 u7 u5 du = − + C cos7x cos5x 75 = − + C. 75 Æèøýý 143. sin3x cos5xdx èíòåãðàëûã áîä. u = sin x, u = cos x. u = cos x, du = − sin xdx I = sin3x cos5xdx = sin2x cos5x sin xdx. sin2 x = 1 − cos2 x, I = sin2x cos5x sin xdx = 1 − cos2x cos5x sin xdx =− 1 − u2 u5du = u7 − u5 u8 u6 du = − + C 86

Ãàð÷èã 101 cos8x cos6x = − + C. 86 Æèøýý 144. sin3x cos3xdx èíòåãðàëûã áîä. u = sin x, u = cos x, u = sin x, du = cos xdx I = sin3x cos3xdx = sin3x cos2x cos xdx sin2x = 1 − cos2x, sin3x 1 − sin2x cos xdx I = sin3x cos2x cos xdx = = u3 1 − u2 du = u3 − u5 du = u4 − u6 + C 46 = sin4x − sin6x + C. 46 Æèøýý 145. sin3x√cos xdx èíòåãðàëûã áîä. u cos x, du = − sin xdx, sin2 x = 1 − cos2 x, 1 − cos2x sin3x√cos xdx = sin2x√cos x sin xdx = √ 1 − u2 √ 1 − 5 du cos x sin xdx = − udu = − u2 u2 u 5 +1 u 1 +1 27 23 2 2 u2 u2 − − = 5 + 1 1 + 1 +C = 7 3 +C 2 2

0.5. Òðèãîíîìåòðèéí èíòåãðàë 102 = 2 √ − 2 √ + C. cos7x cos3x 73 Æèøýý 146. sin 2x cos 5xdx èíòåãðàëûã áîä. sin ax cos bx = 1 [sin (ax + bx) + sin (ax − bx)] . 2 sin 2x cos 5x = 1 [sin 7x + sin (−3x)] = 1 [sin 7x − sin 3x] . 22 1 (sin 7x − sin 3x) dx sin 2x cos 5xdx = 2 1 −cos 7x − −cos 3x +C = cos 3x − cos 7x + C. = 3 6 14 27 Æèøýý 147. xx sin cos dx èíòåãðàëûã áîä. 43 sin ax cos bx = 1 [sin (ax + bx) + sin (ax − bx)] . 2 x x1 xx + sin x−x sin cos = sin + 4 32 43 43 1 7x − sin x . = sin 2 12 12 xx 1 7x − sin x dx sin cos dx = sin 12 12 43 2

Ãàð÷èã 103 1 cos 7x − cos x +C = 6 7x − 6 cos x + C. = 12 12 cos 2 7 1 7 12 12 12 12 Æèøýý 148. xx cos cos dx èíòåãðàëûã áîä. 23 cos ax cos bx = 1 [cos (ax + bx) + cos (ax − bx)] . 2 x x1 cos xx + cos x−x cos cos = + 23 2 32 23 1 5x x = cos + cos . 26 6 xx 1 5x x cos cos dx = cos + cos dx 23 2 66 1 sin 5x sin x 3 5x x = 6 + 6 + C = sin + 3 sin + C. 1 2 5 56 6 66 Æèøýý 149. xx sin sin dx èíòåãðàëûã áîä. 34 sin ax sin bx = −1 [cos (ax + bx) − cos (ax − bx)] , 2 sin x sin x = −1 cos 7x − cos x . 3 4 2 12 12 sin x sin x = −1 7x − cos x dx dx cos 34 2 12 12

0.5. Òðèãîíîìåòðèéí èíòåãðàë 104 1 cos x − cos 7x 1 sin x − sin 7x +C = dx = 12 12 2 12 12 2 1 7 12 12 = 6 sin x − 6 sin 7x + C. 12 7 12 Æèøýý 150. tg4xdx èíòåãðàëûã áîä. 1 + tg2x = sec2x tg4xdx = tg2x tg2xdx = tg2x sec2x − 1 dx = tg2x sec2xdx − tg2xdx = tg2x d (tg x) − sec2x − 1 tg3x dx = − tg x + x + C. 3 Æèøýý 151. ctg4xdx èíòåãðàëûã áîä. ctgnxdx = − ctgn−1x − ctgn−2xdx, n−1 I = ctg4xdx = −ctg3x − ctg2xdx. 3 I = −ctg3x − ctg2xdx = −ctg3x − − ctg x − 1dx 33 = −ctg3x + ctg x + x + C. 3 Æèøýý 152.

Ãàð÷èã 105 tg5 xdx èíòåãðàëûã áîä. tgnxdx = tgn−1x − tgn−2xdx. tg xdx n−1 I = tg5xdx = tg4x − tg3xdx. 4 I = tg4x − tg3xdx = tg4x − tg2x − 4 42 = tg4x − tg2x + tg xdx. 42 tg xdx = − ln |cos x| . I = tg4x − tg2x + tg xdx = tg4x − tg2x − ln |cos x| + C. 42 42 Æèøýý 153. ctg5xdx èíòåãðàëûã áîä. 1 + ctg2x = csc2x, ctg5xdx = ctg3x ctg2xdx = ctg3x csc2x − 1 dx = ctg3x csc2xdx − ctg3xdx = − ctg3x d (ctg x) − ctg x ctg2xdx

0.5. Òðèãîíîìåòðèéí èíòåãðàë 106 ctg4x ctg x csc2x − 1 dx =− − 4 ctg4x ctg x csc2xdx + ctg xdx =− − 4 ctg4x d (sin x) = − − ctg x d (ctg x) + 4 sin x ctg4x ctg2x + ln |sin x| + C. =− + 42 Æèøýý 154. sec3xdx èíòåãðàëûã áîä. secnxdx = secn−2x tg x + n − 2 secn−2xdx. n−1 n−1 sec3xdx = sec x tg x + 1 sec xdx. 22 xπ sec xdx = ln tg + . 24 sec3xdx = sec x tg x + 1 tg xπ + C. ln + 22 24 Æèøýý 155. csc3xdx èíòåãðàëûã áîä. cscnxdx = cscn−2x ctg x + n − 2 cscn−2xdx. − 1 n − 1 n−

Ãàð÷èã 107 I = csc3xdx = −csc x ctg x + 1 csc xdx. 22 I = −csc x ctg x + 1 csc xdx = −csc x ctg x + 1 tg x ln 22 2 22 + C. Æèøýý 156. csc4xdx èíòåãðàëûã áîä. cscnxdx = cscn−2x ctg x + n − 2 cscn−2xdx, − 1 n − 1 n− csc4xdx = −csc2x ctg x + 2 csc2xdx 33 = −csc2x ctg x − 2 ctg x + C 33 = −ctg x csc2x + 2 + C. 3 Æèøýý 157. sec4xdx èíòåãðàëûã áîä. secnxdx = secn−2x tg x + n − 2 secn−2xdx. n−1 n − 1 I = sec4xdx = sec2x tg x + 2 sec2xdx. 33 sec2x tg x 2 sec2xdx = sec2x tg x + 2 tg x + C I= + 33 33

0.5. Òðèãîíîìåòðèéí èíòåãðàë 108 = tg x sec2x + 2 + C. 3 Æèøýý 158. tg3x sec2xdx èíòåãðàëûã áîä. tg3x sec2xdx = tg3x d (tg x) = tg4x + C. 4 Æèøýý 159. tg2x sec xdx èíòåãðàëûã áîä. 1 + tg2x = sec2x. sec2x − 1 sec xdx = sec3xdx I = tg2x sec xdx = − sec xdx. I= sec3xdx − sec xdx = sec x tg x 1 xπ + ln tg + 22 24 − ln tg xπ + C = sec x tg x − 1 ln tg xπ + C. + + 24 22 24