91Ó°ÊÓ

When faced with the integral \( \int_{0}^{\pi / 2} \frac{\cos x \, dx}{1+\sin x} \), we utilize the substitution method to simplify the expression. The idea is to replace a portion of the integral with a new variable, making the integral easier to evaluate.

First, identify a part of the integrand (the function within the integral) that can be substituted. We notice that if we let \( u = 1 + \sin x \), then the differential becomes \( du = \cos x \, dx \). This choice is strategic because it replaces the complicated integrand into a simpler form \( \int \frac{du}{u} \).

Using substitution:

• Choose a substitution that simplifies the integral.
• Differentiate to find the new differential.
• Ensure continuity by transforming the integration limits according to the new variable.

Utilizing substitution effectively can transform an otherwise complicated integral into a standard form, enhancing both accuracy and ease of calculation.

A definite integral has specific upper and lower limits, providing not just a function, but a specific value. In the given exercise, \( \int_{0}^{\pi / 2} \frac{\cos x \, dx}{1+\sin x} \) becomes, through substitution, a definite integral of \( \int_{1}^{2} \frac{du}{u} \).

This integral suggests a standard evaluation between two limits:

• Substitute every occurrence of the original variable with the new variable limits.
• Find the antiderivative using these transformed limits.
• Evaluate the difference of antiderivative values at these boundary limits.

In simple terms, a definite integral calculates the accumulated change, or "area" under a curve between two points. It is precise and provides numerical solutions necessary in various applications such as calculating total distance or predicting consumption over time.

After applying the substitution, the integral \( \int \frac{du}{u} \) represents a standard form seen in logarithmic integrals. Fundamentally, the integral of \( \frac{1}{u} \) with respect to \( u \) is the natural logarithm function, written as \( \ln|u| + C \), where \( C \) is the constant of integration for indefinite integrals.

In definite integrals, like \( \int_{1}^{2} \frac{du}{u} \), the natural logarithm simplifies further because the constant of integration cancels when we calculate difference at bounds:

• Calculate \( \ln(u) \) at the upper and lower limits.
• Subtract the lower limit value from the upper limit value.
• In our exercise, we found \( \ln(2) - \ln(1) \), simplifying to \( \ln(2) \).

This illustrates the utility and simplicity of logarithmic functions in integration, where the change between two values is often expressed as a natural logarithmic difference, offering cohesive and interpretable results.