Find the Indefinite Integral without u substitution of the following
a)(8x-3)/(12x^6)
b) 2sqrtx(5x-6)
c) (4x+3)/xsqrtx
d) (5x-6)^2 / sqrt x
e) 16x^2 + x - 20 / 4x^2
Expand each out, as this:
b. 10 sqrtx*x-12sqrtx
10 x^3/2 - 12 x^1/2
now use the power rule.
c. 4x^-1/2 + 3x^-3/2 use the power rule
d. expand the numerator, then write the sum of the three terms, use the power rule.
To find the indefinite integral of each expression without u-substitution, we can use various integration techniques. Let's go through each expression step by step:
a) Indefinite integral of (8x - 3) / (12x^6):
First, break down the fraction into two separate terms: 8x / (12x^6) - 3 / (12x^6).
Now, let's integrate each term:
∫(8x / (12x^6)) dx = (2/3) * ∫(1 / x^5) dx = (2/3) * (-1/4) * x^(-4) = -x^(-4) / 6x^3 = -1 / 6x^7.
∫(-3 / (12x^6)) dx = -(1/4) * ∫(1 / x^6) dx = -(1/4) * (-1/5) * x^(-5) = x^(-5) / 20x^4 = 1 / 20x^9.
Adding both integrals together, we get the final result:
Indefinite integral of (8x - 3) / (12x^6) = -1 / 6x^7 + 1 / 20x^9 + C, where C is the constant of integration.
b) Indefinite integral of 2√x(5x - 6):
First, distribute the 2√x and write the expression as 2√x * 5x - 2√x * 6.
Now, let's integrate each term:
∫(2√x * 5x) dx = 10 * ∫(x^1/2 * x) dx = 10 * (1/4) * x^(3/2 + 1) = 5/2 * x^(5/2).
∫(2√x * -6) dx = -12 * ∫(√x) dx = -24/3 * x^(3/2) = -8x^(3/2).
Adding both integrals together, we get the final result:
Indefinite integral of 2√x(5x - 6) = 5/2 * x^(5/2) - 8x^(3/2) + C, where C is the constant of integration.
c) Indefinite integral of (4x + 3) / (x√x):
In this case, we can rewrite the expression as (4x / x√x) + (3 / x√x).
Now, let's integrate each term:
∫(4x / x√x) dx = ∫(4 / √x) dx = 8 * ∫(1 / √x) dx = 8 * 2√x = 16√x.
∫(3 / x√x) dx = 6 * ∫(1 / √x) dx = 6 * 2√x = 12√x.
Adding both integrals together, we get the final result:
Indefinite integral of (4x + 3) / (x√x) = 16√x + 12√x + C = 28√x + C, where C is the constant of integration.
d) Indefinite integral of (5x - 6)^2 / √x:
To integrate this expression, we can expand the square term:
(5x - 6)^2 = (5x - 6)(5x - 6) = 25x^2 - 60x + 36.
Now, let's integrate each term:
∫(25x^2 / √x) dx = 25 ∫(x^(3/2)) dx = 25 * (2/5) * x^(5/2) = 10x^(5/2).
∫(-60x / √x) dx = -60 ∫(√x) dx = -60 * (2/3) * x^(3/2) = -40x^(3/2).
∫(36 / √x) dx = 36 ∫(√x) dx = 36 * (2/3) * x^(3/2) = 24x^(3/2).
Adding all three integrals together, we get the final result:
Indefinite integral of (5x - 6)^2 / √x = 10x^(5/2) - 40x^(3/2) + 24x^(3/2) + C = 10x^(5/2) - 16x^(3/2) + C, where C is the constant of integration.
e) Indefinite integral of (16x^2 + x - 20) / (4x^2):
First, break down the fraction into three separate terms: 16x^2 / (4x^2) + x / (4x^2) - 20 / (4x^2).
Now, let's integrate each term:
∫(16x^2 / (4x^2)) dx = 4 * ∫(1) dx = 4x.
∫(x / (4x^2)) dx = (1/4) * ∫(1 / x) dx = (1/4) * ln|x|.
∫(-20 / (4x^2)) dx = -5 * ∫(1 / x^2) dx = -5 * (-1 / x) = 5/x.
Adding all three integrals together, we get the final result:
Indefinite integral of (16x^2 + x - 20) / (4x^2) = 4x + (1/4) * ln|x| + 5/x + C, where C is the constant of integration.