91Ó°ÊÓ

Trigonometric substitution is a powerful technique used to simplify and solve integrals, especially when dealing with expressions involving square roots. In this exercise, we have an integral with a square root in the denominator:

• \( \int \frac{x}{\sqrt{x^{2}+4}} \, dx \)

The presence of \( x^2 + 4 \) under a square root suggests using trigonometric identities. Here, we use the substitution \( x = 2\tan(\theta) \), as it helps transform the integral into a more manageable form.
With \( x = 2\tan(\theta) \), we find that \( dx = 2\sec^2(\theta) \, d\theta \), simplifying the expression under the square root:

• \( \sqrt{x^2 + 4} = \sqrt{4\tan^2(\theta) + 4} = 2\sec(\theta) \)

The integral transforms based on this substitution, paving the way to apply trigonometric identities like \( 1 + \tan^2(\theta) = \sec^2(\theta) \), which simplifies our work greatly.

Although this problem focuses on indefinite integrals, one might wonder how definite integrals come into play with trigonometric substitution. A key aspect is the transformation of limits when substituting variables. Suppose we are solving a definite integral with limits \( a \) to \( b \):

• First, replace the original limits with new limits based on your substitution.

For example, if \( x = 2\tan(\theta) \), we would need to convert both \( a \) and \( b \) into their corresponding \( \theta \) values through \( \theta = \arctan\left(\frac{x}{2}\right) \).
The key takeaway is that integration bounds must also be transformed alongside the variables, ensuring we accurately evaluate the integral.
This ensures a smooth transition and maintains the integrity of the integral within the given interval.

Checking your integration results through differentiation is a crucial step. It confirms the accuracy of your solution.This problem ends with a differentiation step to validate:

• \( -\frac{8}{\sqrt{x^2+4}} + C \)

Use the chain rule to differentiate. When differentiating \( -\frac{8}{\sqrt{x^2+4}} \), apply:

• Derivative of the outer function: \(-8\) times the derivative of \( \frac{1}{\sqrt{x^2+4}} \)
• Chain rule for \( \frac{1}{\sqrt{x^2+4}} \) gives: \( \frac{-x}{(x^2+4)^{3/2}} \)

Finally, combining these components, you recover the original integrand, \( \frac{x}{\sqrt{x^2+4}} \), verifying that your integration was performed correctly.
This method ensures the solution is sound and reinforces the role of differentiation as a powerful tool for verification in calculus.