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

∫x2dxx2−4 \int \frac{x^{2} d x}{x^{2}-4} =

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Full solution

Q. ∫x2dxx2−4 \int \frac{x^{2} d x}{x^{2}-4} =
  1. Simplify Integral Expression: Step 11: Simplify the integral expression.\newlineWe start by noticing that the integral can be simplified by partial fractions, but first, let's check if direct integration is possible.\newlineThe integral is ∫x2x2−4 dx\int\frac{x^2}{x^2 - 4} \, dx.
  2. Attempt Direct Integration: Step 22: Attempt direct integration.\newlineWe attempt to simplify the integrand:\newlinex2x2−4=1+4x2−4\frac{x^2}{x^2 - 4} = 1 + \frac{4}{x^2 - 4}.\newlineNow, we integrate each part separately:\newline∫x2x2−4dx=∫1dx+∫4x2−4dx\int \frac{x^2}{x^2 - 4} dx = \int 1 dx + \int \frac{4}{x^2 - 4} dx.
  3. Integrate First Part: Step 33: Integrate the first part.\newlineThe integral of 11 with respect to xx is xx.\newlineSo, ∫1 dx=x\int 1 \, dx = x.
  4. Integrate Second Part: Step 44: Integrate the second part using partial fractions.\newlineWe decompose 4x2−4\frac{4}{x^2 - 4} into partial fractions:\newline4x2−4=4(x−2)(x+2)=Ax−2+Bx+2\frac{4}{x^2 - 4} = \frac{4}{(x-2)(x+2)} = \frac{A}{x-2} + \frac{B}{x+2}.\newlineSolving for AA and BB, we find A=1A = 1, B=1B = 1.\newlineSo, 4x2−4=1x−2+1x+2\frac{4}{x^2 - 4} = \frac{1}{x-2} + \frac{1}{x+2}.\newlineNow, integrate:\newline∫4x2−4dx=∫(1x−2+1x+2)dx=ln⁡∣x−2∣+ln⁡∣x+2∣+C\int \frac{4}{x^2 - 4} dx = \int (\frac{1}{x-2} + \frac{1}{x+2}) dx = \ln|x-2| + \ln|x+2| + C.
  5. Combine Results: Step 55: Combine the results.\newlineThe integral of the original function is:\newline∫x2x2−4dx=x+ln⁡∣x−2∣+ln⁡∣x+2∣+C\int\frac{x^2}{x^2 - 4} dx = x + \ln|x-2| + \ln|x+2| + C.

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