Solving the Integral ∫ x1/2 / (x2 1) dx Step-by-Step

This article provides a detailed step-by-step guide on how to solve the integral ∫ x1/2 / (x2 1) dx. Let's break down the process into manageable parts, making use of integral calculus techniques such as substitution.

Introduction to the Problem

The given integral is:

(I displaystyle int frac{sqrt{x1}}{x^2 1} dx)

Substitution and Simplification

The first step in solving this integral is to use a suitable substitution. Let's choose:

(x^1 t^2)

This implies:

(dx 2t dt)

Substituting these values into the original integral:

(I displaystyle int frac{t}{t^2 - 1^2 1} 2t dt)

Simplifying the integrand:

(I displaystyle int frac{2t^2}{t^4 - 2t^2 2} dt)

Further Simplification

Next, we simplify the integrand further:

(I displaystyle int frac{2}{t^2 - 2 frac{2}{t^2}} dt)

This can be rearranged as:

(I displaystyle int frac{1 frac{sqrt{2}}{t^2} - 1 - frac{sqrt{2}}{t^2}}{t^2 - 2 frac{2}{t^2}} dt)

Which breaks down into:

(I displaystyle int frac{1 frac{sqrt{2}}{t^2}}{t^2 - 2sqrt{2} 1} dt displaystyle int frac{1 - frac{sqrt{2}}{t^2}}{t^2 - 2sqrt{2} - 1} dt)

Solving Each Integral Separately

To solve these integrals, we use the standard forms of integrals involving quadratic expressions:

(int frac{dx}{x^2 a^2} frac{1}{a}tan^{-1}left(frac{x}{a}right) C)

(int frac{dx}{x^2 - a^2} frac{1}{2a}lnleft|frac{x - a}{x a}right| C)

Applying these formulas to our integrals:

(I frac{1}{sqrt{2sqrt{2} 1}}tan^{-1}left(frac{t - frac{sqrt{2}}{t}}{sqrt{2sqrt{2} - 1}}right) - frac{1}{2sqrt{2sqrt{2} - 1}}lnleft|frac{t - frac{sqrt{2}}{t} - sqrt{2sqrt{2} - 1}}{t - frac{sqrt{2}}{t} sqrt{2sqrt{2} - 1}}right| C)

Finally, substituting back (t^2 x1):

(I frac{1}{sqrt{2sqrt{2} 1}}tan^{-1}left(frac{t^2 - sqrt{2}}{tsqrt{2sqrt{2} - 1}}right) - frac{1}{2sqrt{2sqrt{2} - 1}}lnleft|frac{t^2 - sqrt{2} - sqrt{2sqrt{2} - 1}}{t^2 - sqrt{2} sqrt{2sqrt{2} - 1}}right| C)

Conclusion

Thus, the solution to the given integral is:

(I frac{1}{sqrt{2sqrt{2} 1}}tan^{-1}left(frac{x1 - sqrt{2}}{sqrt{2sqrt{2} - 1} cdot sqrt{x1}}right) - frac{1}{2sqrt{2sqrt{2} - 1}}lnleft|frac{x1 - sqrt{2} - sqrt{2sqrt{2} - 1}}{x1 - sqrt{2} sqrt{2sqrt{2} - 1}}right| C)

where (C) is the integration constant.

This step-by-step approach demonstrates the power of substitution and standard integral formulas. For further practice or understanding, you might want to try similar problems and explore other substitution techniques in calculus.

Final Answer:

(boxed{I frac{1}{sqrt{2sqrt{2} 1}}tan^{-1}left(frac{x1 - sqrt{2}}{sqrt{2sqrt{2} - 1} cdot sqrt{x1}}right) - frac{1}{2sqrt{2sqrt{2} - 1}}lnleft|frac{x1 - sqrt{2} - sqrt{2sqrt{2} - 1}}{x1 - sqrt{2} sqrt{2sqrt{2} - 1}}right| C})