Calculating the Integral of dx/{(x-1)(x^2+4)^1/2}
- Partial Fraction Decomposition: First, express the given expression as a sum of simpler fractions by performing partial fraction decomposition. Let's assume the integral can be expressed as A/(x-1) + Bx/(x^2+4).
- Find A and B: By equating coefficients, we find that A = 1/4 and B = -1/4.
- Integral Calculation: Now, we can rewrite the integral as ∫(1/(4(x-1)) - x/(4(x^2+4))) dx.
- Integrate Each Term: Integrate each term separately. The integral of 1/(4(x-1)) is ln|x-1|/4 and the integral of -x/(4(x^2+4)) can be found using a trigonometric substitution.
- Final Answer: Putting it all together, the final answer is (1/4)ln|x-1| - (1/8)arctan(x/2) + C, where C is the constant of integration.
- Conclusion: By following the steps of partial fraction decomposition and integrating each term separately, we were able to calculate the integral of dx/{(x-1)(x^2+4)^1/2}.

myself new..i could not understand sign use here