∫ (3-2x) ^ 3DX for indefinite integral

∫ (3-2x) ^ 3DX for indefinite integral

Four methods of temporary post: quick calculation of ∫ (3 - 2x)? DX
= 1 / 2 * ∫ (3 - 2 x)? d(2 x)
= - 1 / 2 * ∫ (3 - 2 x)? d(- 2 x)
= - 1 / 2 * ∫ (3 - 2 x)? d(3 - 2 x)
= - 1 / 2 * (3 - 2 x)^4 / 4 + C
=[- (3-2x) ^ 4] / 8 + C method 2: substitution ∫ (3-2x)? DX
Let u = 3 - 2 X
du = - 2 dx
dx = - du / 2
The original formula = - 1 / 2 ∫ u? Du
= - 1 / 2 * u^4 / 4 + C
=[- (3-2x) ^ 4] / 8 + C method 3: binomial expansion ∫ (3-2x)? DX
= ∫ [3? - 3 (3?) (2 x) + 3 (3） (2 x)? - (2 x)?] dx
= ∫ (27 - 54 x + 36 x? - 8 x?) dx
= 27 x - 54 x? / 2 + 36 x? / 3 - 8 x^4 / 4 + C
=27 x - 27 x? + 12 x? - 2 x ^ 4 + C plus (- 81 / 8) factorization to get = [- (3 - 2 x) ^ 4] / 8 + C 'method 4: trigonometric function substitution ∫ (3 - 2 x)? DX
Let √ (2 x) = √ 3 sin θ
Sin? θ = √ (2 x) / √ 3 with cos? θ = √ (1 - sin? θ) formula
Cos θ = √ (3 - 2 x) / √ 3cos θ
x = 3 sin?θ / 2
dx = 3 sinθ cosθ dθ
(3 - 2 x)?=(3 - 2 * 3 sin?θ / 2)?
=(3 cos? θ)? = 27 [cosθ]^6
The original formula = ∫ 27 [cos θ] ^ 6 * 3 sin θ cos θ D θ
= - 81 ∫ [cosθ]^7 d(cosθ)
= - 81 * [cosθ]^8 / 8 + C
= - 81 / 8 * [√(3 - 2 x) / √3]^8 + C
= - 81 / 8 * (3 - 2 x)^(1 / 2 * 8) / 3^(1 / 2 * 8) + C
= [- (3 - 2 x)^4] / 8 + C

How to find the indefinite integral of ∫ x √ x + 1dx?
How to find the indefinite integral of X multiplied by (x + 1) to the power of 1 / 2?

Let t = √ (x + 1)
Then x = T ^ 2-1
dx=2tdt;
∫x√x+1dx=∫2t^2(t^2-1)dt
=∫(2t^4-2t^2)dt
=(2/5)t^5-(2/3)t^3+C
From t = √ (x + 1)
=(2/5)(x+1)^(5/2)-(2/3)(x+1)^(3/2)+C

Seeking indefinite integral ∫ (2x-3) / (x ^ 2 - 2x + 2)

(x² - 2x + 2)' = 2x - 2,2x - 3 = (2x - 2) - 1
∫ (2x - 3)/(x² - 2x + 2) dx
= ∫ [(2x - 2) - 1]/(x² - 2x + 2) dx
= ∫ (2x - 2)/(x² - 2x + 2) dx - ∫ dx/(x² - 2x + 2)
= ∫ d(x² - 2x + 2)/(x² - 2x + 2) - ∫ d(x - 1)/[(x - 1)² + 1],【∫ dx/(x² + a²) = (1/a)arctan(x/a) + C】
= ln[(x - 1)² + 1] - arctan(x - 1) + C

Indefinite integral of X / (x ^ 2 + 2x-3) ^ 1 / 2

∫ x/√(x² + 2x - 3) dx= ∫ x/√[(x + 1)² - 4] dx,x + 1 = 2secz,dx = 2secztanz dz= ∫ (2secz - 1)/|2tanz| * (2secztanz dz)= ∫ (2secz - 1) * secz dz= 2∫ (sec²z - secz) dz= 2tanz - ln|s...