int(x^(2)dx)/(x^(2)-4)=

Simplify Integral Expression: Step 1: Simplify the integral expression. We start by noticing that the integral can be simplified by partial fractions, but first, let's check if direct integration is possible. The integral is ∫(x^2)/(x^2 - 4) dx.Attempt Direct Integration: Step 2: Attempt direct integration. We attempt to simplify the integrand: (x^2)/(x^2 - 4) = 1 + 4/(x^2 - 4). Now, we integrate each part separately: ∫(x^2)/(x^2 - 4) dx = ∫1 dx + ∫4/(x^2 - 4) dx.Integrate First Part: Step 3: Integrate the first part. The integral of 1 with respect to x is x. So, ∫1 dx = x.Integrate Second Part: Step 4: Integrate the second part using partial fractions. We decompose 4/(x^2 - 4) into partial fractions: 4/(x^2 - 4) = 4/((x-2)(x+2)) = A/(x-2) + B/(x+2). Solving for A and B, we find A = 1, B = 1. So, 4/(x^2 - 4) = 1/(x-2) + 1/(x+2). Now, integrate: ∫4/(x^2 - 4) dx = ∫(1/(x-2) + 1/(x+2)) dx = ln|x-2| + ln|x+2| + C.Combine Results: Step 5: Combine the results. The integral of the original function is: ∫(x^2)/(x^2 - 4) dx = x + ln|x-2| + ln|x+2| + C.

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$\int \frac{x^{2} d x}{x^{2}-4}$ =

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Calculus

Evaluate definite integrals using the chain rule

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\int \frac{x^{2} d x}{x^{2}-4}

Simplify Integral Expression: Step $1$ : Simplify the integral expression. $\newline$ We start by noticing that the integral can be simplified by partial fractions, but first, let's check if direct integration is possible. $\newline$ The integral is $\int\frac{x^2}{x^2 - 4} \, dx$ .
Attempt Direct Integration: Step $2$ : Attempt direct integration. $\newline$ We attempt to simplify the integrand: $\newline$ $\frac{x^2}{x^2 - 4} = 1 + \frac{4}{x^2 - 4}$ . $\newline$ Now, we integrate each part separately: $\newline$ $\int \frac{x^2}{x^2 - 4} dx = \int 1 dx + \int \frac{4}{x^2 - 4} dx$ .
Integrate First Part: Step $3$ : Integrate the first part. $\newline$ The integral of $1$ with respect to $x$ is $x$ . $\newline$ So, $\int 1 \, dx = x$ .
Integrate Second Part: Step $4$ : Integrate the second part using partial fractions. $\newline$ We decompose $\frac{4}{x^2 - 4}$ into partial fractions: $\newline$ $\frac{4}{x^2 - 4} = \frac{4}{(x-2)(x+2)} = \frac{A}{x-2} + \frac{B}{x+2}$ . $\newline$ Solving for $A$ and $B$ , we find $A = 1$ , $B = 1$ . $\newline$ So, $\frac{4}{x^2 - 4} = \frac{1}{x-2} + \frac{1}{x+2}$ . $\newline$ Now, integrate: $\newline$ $\int \frac{4}{x^2 - 4} dx = \int (\frac{1}{x-2} + \frac{1}{x+2}) dx = \ln|x-2| + \ln|x+2| + C$ .
Combine Results: Step $5$ : Combine the results. $\newline$ The integral of the original function is: $\newline$ $\int\frac{x^2}{x^2 - 4} dx = x + \ln|x-2| + \ln|x+2| + C$ .