Solving Complex Integrals: A Detailed Guide to Integration Techniques

Welcome to our detailed guide on solving complex integrals. In this article, we will break down the process of integrating a specific function involving polynomials and square roots, and explore various integration techniques to achieve the solution.

Understanding the Integral

The given integral, which is part of a mathematical problem, involves a combination of polynomial terms and a square root term. To solve this, we will use a series of algebraic manipulations and substitutions, ultimately leading to a solution that can be understood and applied to similar problems.

The Given Integral

The integral we are dealing with is as follows:

[int frac{x^2 2x 3}{x 2sqrt{x^2 1}} , dx]

Step 1: Break Down the Integral

The first step is to break down the integrand into more manageable parts by using algebraic manipulation. We rewrite the numerator using an identity:

[frac{x^2 2x 3}{x 2sqrt{x^2 1}} frac{(x 2)^2 - 2x - 1}{x 2sqrt{x^2 1}}]

Step 2: Split the Integral

Now we split the integral into two parts:

[int frac{(x 2)^2 - 2x - 1}{x 2sqrt{x^2 1}} , dx int frac{(x 2)^2}{x 2sqrt{x^2 1}} , dx - int frac{2x 1}{x 2sqrt{x^2 1}} , dx]

Step 3: Further Manipulation

We continue by splitting and manipulating the integrals further:

[ int frac{x 2}{sqrt{x^2 1}} , dx - int frac{2x 1}{x 2sqrt{x^2 1}} , dx]

[ int frac{x 2}{sqrt{x^2 1}} , dx - left(2int frac{1}{sqrt{x^2 1}} , dx - int frac{2x 2 - 3}{x 2sqrt{x^2 1}} , dxright)]

[ int frac{x 2}{sqrt{x^2 1}} , dx - 2int frac{1}{sqrt{x^2 1}} , dx 2int frac{1}{sqrt{x^2 1}} , dx - 3int frac{1}{x 2sqrt{x^2 1}} , dx]

[ int frac{x 2}{sqrt{x^2 1}} , dx - 3int frac{1}{x 2sqrt{x^2 1}} , dx]

[ sqrt{x^2 1} - 3int frac{1}{x 2sqrt{x^2 1}} , dx]

Step 4: Substitution

To solve the remaining integral, we use the substitution (x sinh(theta)). This substitution helps simplify the integrand:

[int frac{1}{x 2sqrt{x^2 1}} , dx int frac{1}{sinh(theta) 2sqrt{sinh^2(theta) 1}} cosh(theta) , dtheta]

[ int frac{1}{sinh(theta) 2sqrt{cosh^2(theta)}} cosh(theta) , dtheta]

[ int frac{1}{sinh(theta) 2} , dtheta]

[ int frac{1}{left(frac{e^theta - e^{-theta}}{2} 2right)} , dtheta]

[ int frac{2}{e^{2theta} - 1 4e^theta} , dtheta]

[ 2int frac{e^theta}{e^{2theta} - 1 4e^theta} , dtheta]

[ 2int frac{e^theta}{(e^theta)^2 - (2)^2 (sqrt{5})^2} , dtheta]

[ frac{2}{2sqrt{5}}lnleft(frac{e^theta 2 - sqrt{5}}{e^theta 2 sqrt{5}}right) C]

[ frac{1}{sqrt{5}}lnleft(frac{sinh(theta) cosh(theta) 2 - sqrt{5}}{sinh(theta) cosh(theta) 2 sqrt{5}}right) C]

[ frac{1}{sqrt{5}}lnleft(frac{x sqrt{x^2 1} 2 - sqrt{5}}{x sqrt{x^2 1} 2 sqrt{5}}right) C]

Final Result

Putting this result back into the previous equation, we get:

[int frac{x^2 2x 3}{x 2sqrt{x^2 1}} , dx sqrt{x^2 1} - frac{3}{sqrt{5}}lnleft(frac{x sqrt{x^2 1} 2 - sqrt{5}}{x sqrt{x^2 1} 2 sqrt{5}}right) C]

Conclusion

This comprehensive guide has provided a step-by-step approach to solving a complex integral involving polynomials and square roots. We have used algebraic manipulation, substitution, and logarithmic integration to find the final result. These techniques are invaluable for solving similar problems and can be adapted in various mathematical contexts.

Keywords

Complex Integrals

Integrals involving more complex functions can be challenging but are crucial in many fields of mathematics and its applications.

Integration Techniques

Mastering various integration techniques is essential for solving a wide range of mathematical problems, from pure mathematics to applied sciences.

Mathematical Integration

Understanding and applying integration methods is fundamental in calculus and advanced mathematics.