A- maths(Integration)問題2

1. ∫ x √(2x) dx = √(2) ∫ x^(3/2) dx
= √(2) x^(5/2) *(2/5)+C = (2√2)/5 * x^(5/2) +C

2. ∫ 2 ln 3x dx = 2 ∫ ln 3x dx
= 2 [x ln 3x - ∫ x d (ln 3x)]
= 2 [x ln 3x - ∫ x (1/(3x)) dx]
=2 [x ln 3x - ∫ 1/3 dx]
=2 [x ln 3x -x/3] +C

3. let u=1+2x, du = 2dx
when x = 1, u=3 ; when u=2. u= 5
Intergal = ∫ [3~5] u^(-4) (1/2) du
= (1/2) [(1/-3) u^(-3)]| [3~5]
= (-1/6) (1/125 - 1/27) = 49/10125

4. Let u=x^4 +2x^3-x^2 +x+2
du/dx = 4x^3+6x^2-2x+1 =>du = (4x^3+6x^2-2x+1 ) dx
Integral = ∫ du/u = ln |u| +C = ln |x^4 +2x^3-x^2 +x+2| +C

5. x^2 -x-12 = (x+3) (x-4)
Let (5x+1)/(x^2 -x-12) = A/(x+3) +B/(x-4)
5x+1 = A(x-4) + B(x+3) = (A+B) x + (-4A+3B)
Hence A+B = 5, 3B-4A =1
A=2, B=3
Integral = ∫ 2/(x+3) +3/(x-4) dx = 2 ln |x+3| + 3 ln |x-4| +C