91Ó°ÊÓ

The substitution method is a powerful tool in solving integrals, particularly those involving complex expressions.

• Identify parts of the integral that resemble a derivative of a function.
• Use these parts to propose a substitution that simplifies the integral.

In this exercise, we are given the integral \( \int \frac{\sec y \tan y}{2+\sec y} \ dy \). The trick here is to notice the relationship between the numerator \( \sec y \tan y \) and the derivative of \( \sec y \). By setting \( u = 2 + \sec y \), you encapsulate most of the complexity into \( u \), simplifying the integral. The derivative \( du = \sec y \tan y \ dy \) fits perfectly, turning the integral into a simpler form, \( \int \frac{1}{u} \ du \). This strategic switch drastically reduces the problem's difficulty, making it solvable using basic integration techniques.Through substitution, we effectively transform the problem into something familiar and manageable, allowing us to solve it with ease.

Logarithmic integration is involved when dealing with integrals that take the form \( \int \frac{1}{u} \, du \). These integrals result in a logarithmic function after integration.

• The integral \( \int \frac{1}{u} \, du \) can be directly integrated to \( \ln |u| + C \), where \( C \) is the constant of integration.
• This result is derived from the definition of the natural logarithm as the antiderivative of \( 1/u \).

In this situation, after performing the substitution in the given problem, we reduce the complex trigonometric integral into \( \int \frac{1}{u} \, du \). This type of integral is straightforward. We remember that after integrating \( \frac{1}{u} \), the solution is \( \ln |u| + C \). This highlights the ease of handling trigonometric integrals once they are transformed into this standard form.

Verification of an integral solution is crucial to ensure accuracy and correctness.

• This process involves differentiating the obtained solution to check if it returns to the original integrand.
• If differentiating the result gives you back the original function inside the integral, the solution is verified as correct.

For our integral \( \int \frac{\sec y \tan y}{2+\sec y} \, dy \), once we obtained the integral result \( \ln |2 + \sec y| + C \), we then differentiate this expression with respect to \( y \).The derivative of \( \ln |2 + \sec y| \) correctly returns us to the initial integrand, \( \frac{\sec y \tan y}{2+\sec y} \). This step confirms that the approach and calculations are flawless, giving you confidence that our integration process is valid and exact. Verifying solutions is one of the most satisfying skills in mathematics, providing reassurance that the work is correct.