ИНТЕГРАЛЫ: 1) ∫(3x^2 - 2√x + 4x - 5) dx 2) ∫√(1 - x) dx 3) ∫ ctg3x dx 4) ∫x^2 dx/ (x^3 -...

Question

ИНТЕГРАЛЫ:
1) ∫(3x^2 - 2√x + 4x - 5) dx
2) ∫√(1 - x) dx
3) ∫ ctg3x dx
4) ∫x^2 dx/ (x^3 - 1)^2

Answer 1

1. S(3x^2 - 2sqrt(x) + 4x - 5)dx = 3x^3/3 - 2 x^(3/2) * 2/3 + 4x^2/2- 5x + c = x^3 - 4/3 * x^(3/2) + 2x^2 - 5x + c

2. S(sqrt(1-x))dx = -S(sqrt(1-x))d(1-x) = - (1-x)^(3/2) * 2/3 + c = -2/3 (1-x)^(3/2)+c

3. S ctg3x dx = 1/3 S(cos3x/sin3x) d(3x) = 1/3 S 1/sin3x d(sin3x) =
1/3 ln(sin3x) + c

4. S x^2 / (x^3-1)^2 dx = 1/3 S 1/(x^3-1)^2 d(x^3-1) = -1/3 (x^3-1)^(-1)+c
= -1/3 * 1/(x^3-1) + c