91Ó°ÊÓ

When tackling more complex integrals, such as those involving trigonometric functions, u-substitution is a powerful technique to simplify the process. The idea is to change the variable in the integral to make it easier to evaluate. In this case, the substitution used was to let \( u = \sqrt{\tan x} \), transforming the integral into a more manageable form.

After choosing the substitution, we express \( u^2 = \tan x \) and differentiate with respect to \( x \) using the chain rule. This allows us to express \( dx \) in terms of \( du \), making the substitution process complete:

\[ dx = \frac{2u}{\sec^2 x} du \]
This transformation is essential because it converts the original problem into an integral entirely in terms of \( u \), aiding in simplification and integration.

Trigonometric identities are fundamental in rewriting complex trigonometric expressions. In this particular problem, to solve the integral \( \int \sqrt{\tan x} \sec^4 x \, dx \), we made use of the identity \( \sec^2 x = 1 + \tan x \).

Using this identity helps us convert expressions involving \( \sec^4 x \) into terms of \( u \), which is derived from \( \tan x \). By realizing that \( \sec^4 x = (1 + u^2)^2 \), we were able to simplify the integration process:

- Replace \( \sqrt{\tan x} \) with \( u \)
- Express \( \sec^4 x \) as \( (1 + u^2)^2 \)
- Integrate with respect to \( u \) instead of \( x \)

These trigonometric identities not only simplify our expressions but also make the problem more approachable from an integration standpoint.

Integration techniques are strategies used to solve integrals that don't initially present with standard forms. In this exercise, the integration technique involved was simplifying the transformed integral from the u-substitution step.

The integral, after substitution and simplification becomes:

\[ 2\int u^2(1 + u^2) \, du \]
This required recognizing the polynomial inside the integral, \( u^2(1 + u^2) \) as \( u^2 + u^4 \), which can be integrated term by term using the power rule:

- Integrate \( u^2 \), giving \( \frac{1}{3}u^3 \)
- Integrate \( u^4 \), giving \( \frac{1}{5}u^5 \)

After integrating, we substitute back the original variable to form the complete solution. This method of integrating polynomials shows how breaking the problem into smaller parts makes the task manageable. In conclusion, understanding various integration techniques gives us a toolkit to solve different types of integrals efficiently.