Integral of sqrt ( 4 - x^2) / x dx

 

Calculus often presents challenges, especially when dealing with intricate integrals. One such challenge arises in the form of integrals involving square roots of quadratic expressions. To conquer these integrals, mathematicians employ various techniques, one of which is trigonometric substitution. In this article, we will explore the concept of trigonometric substitution by solving a specific integral:

$\int \frac{\sqrt{4-{�}^2}}{�}��$. This technique showcases the elegance and power of trigonometric functions in simplifying complex mathematical problems.

Understanding the Problem

The given integral, $\int \frac{\sqrt{4-{�}^2}}{�}��$, seems daunting at first glance due to its complex structure. Trigonometric substitution provides an elegant solution by transforming the integral into a more manageable form.

Trigonometric Substitution: A Clever Approach

Let's initiate the process by employing a trigonometric substitution. We set $�=2\sin (�)$, which allows us to rewrite $��=2\cos (�)��$. Simultaneously, $4-{�}^2$ transforms into $4{\cos }^2(�)$ using the Pythagorean identity ${\sin }^2(�)+{\cos }^2(�)=1$. This substitution simplifies the integral significantly.

Solving the Integral

The integral now becomes $\int \frac{2{\cos }^2(�)}{\sin (�)}��$. By utilizing the identity ${\cos }^2(�)=1-{\sin }^2(�)$, we further simplify the integral to $\int (2\csc (�)-2\sin (�))��$.

Integrating term by term, we arrive at the following expression:

$$-2\ln \mathrm{\mid }\csc (�)+\cot (�)\mathrm{\mid }-2\sin (�)+�,$$

where $�$ represents the constant of integration.

Back Substitution: Reintroducing $�$

To obtain the final solution in terms of $�$, we need to reintroduce $�$ into our expression. Recall that our initial substitution was $�=2\sin (�)$. Using this relationship, we can express $\sin (�)$ in terms of $�$. Substituting back, we arrive at the solution:

−2ln∣csc(θ)+cot(θ)∣+4−x2​+C

Conclusion: Trigonometric Substitution Unveiled

Through the lens of the given integral, we've delved into the technique of trigonometric substitution, unraveling its effectiveness in simplifying intricate problems. By cleverly selecting appropriate trigonometric functions, mathematicians can transform complex integrals into manageable forms, showcasing the elegance and power of mathematical techniques. Trigonometric substitution stands as a testament to the ingenuity inherent in mathematics, offering a glimpse into the creative problem-solving methods employed by mathematicians worldwide.