Use the Substitution Formula in Theorem 7 to evaluate the...

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises $1-24 .$
a. $\int_{0}^{2 \pi} \frac{\cos z}{\sqrt{4+3 \sin z}} d z \quad$ b. $\int_{-\pi}^{\pi} \frac{\cos z}{\sqrt{4+3 \sin z}} d z$

Key Concepts

Properties of Trigonometric Functions

Trigonometric functions have specific periodic and symmetry properties that can simplify integration. Understanding these properties, such as the periodicity of sine and cosine, allows one to recognize patterns and sometimes reduce the range of integration or infer that certain integrals evaluate to zero.

Symmetry in Integration

Symmetry in integration refers to the behavior of functions over symmetric intervals, particularly when dealing with even or odd functions. For example, integrating an odd function over an interval symmetric about zero yields zero, while even functions can often be simplified by focusing on half the domain. This concept is especially useful when the integrals involve trigonometric functions.

Transformation of Integration Limits

When performing substitution in definite integrals, it is crucial to also change the limits of integration. After substituting, the original limits, defined in terms of the original variable, must be recalculated based on the new variable, ensuring that the integral’s evaluation is consistent within the new variable's domain.

Substitution Rule

The substitution rule in integration, often called u?substitution, is a method used to evaluate integrals by changing variables to simplify the integrand. This technique involves choosing a substitution that transforms the integral into one with an easier integrand, essentially reversing the chain rule from differentiation.

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