int((1-sqrt(3x))^(2))/(x)dx

Simplify integrand: Simplify the integrand. We have the integral of (1 - sqrt(3x))^2 / x. Let's expand the square and simplify the expression before integrating. (1 - sqrt(3x))^2 = 1 - 2*sqrt(3x) + 3x Now, divide each term by x: (1 - sqrt(3x))^2 / x = 1/x - 2*sqrt(3)/x^(3/2) + 3Split into three integrals: Split the integral into three separate integrals. We can now write the integral as the sum of three separate integrals: ∫((1 - sqrt(3x))^2 / x) dx = ∫(1/x) dx - 2*sqrt(3)∫(1/x^(3/2)) dx + ∫3 dxIntegrate each term: Integrate each term separately. The first integral is the natural logarithm: ∫(1/x) dx = ln|x| + C1 The second integral involves a power of x: ∫(1/x^(3/2)) dx = ∫x^(-3/2) dx = (x^(-1/2)) / (-1/2) + C2 = -2/x^(1/2) + C2 The third integral is straightforward: ∫3 dx = 3x + C3Combine and simplify: Combine the results and simplify. Combining the results from Step 3, we get: ∫((1 - sqrt(3x))^2 / x) dx = ln|x| - 2*sqrt(3)(-2/x^(1/2)) + 3x + C Simplify the second term: = ln|x| + 4*sqrt(3)/sqrt(x) + 3x + CWrite final answer: Write the final answer. The indefinite integral of (1 - sqrt(3x))^2 / x with respect to x is: ∫((1 - sqrt(3x))^2 / x) dx = ln|x| + 4*sqrt(3)/sqrt(x) + 3x + C

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$\int(\frac{(1-\sqrt{3x})^{2}}{x} dx$

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Evaluate definite integrals using the chain rule

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\int(\frac{(1-\sqrt{3x})^{2}}{x} dx

Simplify integrand: Simplify the integrand. $\newline$ We have the integral of $(1 - \sqrt{3x})^2 / x$ . Let's expand the square and simplify the expression before integrating. $\newline$ $(1 - \sqrt{3x})^2 = 1 - 2\sqrt{3x} + 3x$ $\newline$ Now, divide each term by $x$ : $\newline$ $(1 - \sqrt{3x})^2 / x = 1/x - 2\sqrt{3}/x^{3/2} + 3$
Split into three integrals: Split the integral into three separate integrals. $\newline$ We can now write the integral as the sum of three separate integrals: $\newline$ \int\left(\frac{($1$ - \sqrt{$3$x})^$2$}{x}\right) dx = \int\left(\frac{$1$}{x}\right) dx - \(2\sqrt{ $3$ }\int\left(\frac{ $1$ }{x^{\frac{ $3$ }{ $2$ }}}\right) dx + \int $3$ dx
Integrate each term: Integrate each term separately. $\newline$ The first integral is the natural logarithm: $\newline$ $\int(\frac{1}{x}) dx = \ln|x| + C_1$ $\newline$ The second integral involves a power of x: $\newline$ $\int(\frac{1}{x^{3/2}}) dx = \int x^{-3/2} dx = \frac{x^{-1/2}}{-1/2} + C_2 = -2x^{-1/2} + C_2$ $\newline$ The third integral is straightforward: $\newline$ $\int 3 dx = 3x + C_3$
Combine and simplify: Combine the results and simplify. $\newline$ Combining the results from Step $3$ , we get: $\newline$ $\int\left(\frac{(1 - \sqrt{3x})^2}{x}\right) dx = \ln|x| - 2\sqrt{3}(-2/x^{1/2}) + 3x + C$ $\newline$ Simplify the second term: $\newline$ $= \ln|x| + 4\sqrt{3}/\sqrt{x} + 3x + C$
Write final answer: Write the final answer. $\newline$ The indefinite integral of $(1 - \sqrt{3x})^2 / x$ with respect to $x$ is: $\newline$ $\int\left(\frac{(1 - \sqrt{3x})^2}{x}\right) dx = \ln|x| + 4\cdot\sqrt{3}/\sqrt{x} + 3x + C$