(39) Z 1 ax 2 + bx + c dx = 1 a ln 2ax + b +2 p a(ax 2 + bx + c) (40) Z x ax 2 + bx + c dx = 1 a ax 2 + bx + c- b 2a 3/2 ln 2ax + b +2 p a(ax 2 + bx + c) (41) Z dx (a 2 + x 2 ) 3/2 = x a 2 a 2 + x 2 Integrals with Logarithms (42) Z ln ax dx = x ln ax - x (43) Z x ln x dx = 1 2 x 2 ln x - x 2 4 (44) Z x 2 ln x dx = 1 3 x 3 ln x - x 3 9 (45) Z x n ln x dx = x n+1 ln x n +1 - 1 (n + 1) 2 , n 6= -1 (46) Z ln ax x dx = 1 2 (ln ax) 2 (47) Z ln x x 2 dx = - 1 x - ln x x 5
(48) Z ln(ax + b) dx = x + b a ln(ax + b) - x, a 6=0 (49) Z ln(x 2 + a 2 ) dx = x ln(x 2 + a 2 )+2a tan -1 x a - 2x (50) Z ln(x 2 - a 2 ) dx = x ln(x 2 - a 2 )+ a ln x + a x - a - 2x (51) Z ln ( ax 2 + bx + c ) dx = 1 a 4ac - b 2 tan -1 2ax + b 4ac - b 2 -2x+ b 2a + x ln ( ax 2 + bx + c ) (52) Z x ln(ax + b) dx = bx 2a - 1 4 x 2 + 1 2 x 2 - b 2 a 2 ln(ax + b) (53) Z x ln ( a 2 - b 2 x 2 ) dx = - 1 2 x 2 + 1 2 x 2 - a 2 b 2 ln ( a 2 - b 2 x 2 ) (54) Z (ln x) 2 dx =2x - 2x ln x + x(ln x) 2 (55) Z (ln x) 3 dx = -6x + x(ln x) 3 - 3x(ln x) 2 +6x ln x (56) Z x(ln x) 2 dx = x 2 4 + 1 2 x 2 (ln x) 2 - 1 2 x 2 ln x (57) Z x 2 (ln x) 2 dx = 2x 3 27 + 1 3 x 3 (ln x) 2 - 2 9 x 3 ln x 6
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