If \(\sin (x+y)(d y / d x)=5\) then (A) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}+\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (B) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}-\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (C) \(\int[(\mathrm{dt}) /(5+\operatorname{cosec} \mathrm{t})]=\mathrm{d} \mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (D) \(\int[(\mathrm{dt}) /(5 \sin t+1)]=\mathrm{dt}(\) where \(t=x+y)\)

Understanding various integration techniques is critical for students delving into Integral Calculus. These techniques allow for the systematic approach to solving integrals, which are, in essence, the reverse operation of differentiation. In the textbook exercise, the integral presented is $$5\int \frac{dt}{5+\sin t}$$
To approach this, a student must first identify the type of integral they are dealing with.

There are several key techniques used in integration:

• Substitution: Involves substituting part of the integral with a new variable to simplify the expression.
• Integration by Parts: Based on the product rule for differentiation and is used for integrating products of functions.
• Partial Fractions: Breaks down a complex fraction into simpler parts to integrate.
• Trigonometric Integration: Used when integrating functions involving trigonometric functions.

In our exercise, combining substitution and knowledge of trigonometric integrals can help facilitate the solution.

The technique of trigonometric substitution is particularly handy when dealing with integrals that contain square roots or when trigonometric identities can simplify the integrand. It takes advantage of the properties of trigonometric functions to transform the integral into a form that is more manageable.

For example, when you encounter an integrand that is a function of $$\sin t$$or $$\cos t$$
, you can often employ trigonometric identities to simplify it. In our exercise, we see that $$\sin(x+y)$$is involved, which suggests the opportunity to use trigonometric substitution if necessary. Specifically, you'd look to replace terms using known identities such as $$\sin^2 t + \cos^2 t = 1$$
to transform the integral into a form where standard antiderivatives apply. In the solution provided, while we did not use a direct trigonometric substitution, we capitalized on the relationship between $$\sin t$$and the differential $$dy$$to simplify the integral.

A differential equation is an equation that relates a function to its derivatives. In many branches of science and engineering, modeling scenarios using differential equations is common because it captures the essence of how quantities change in relation to each other. In this textbook exercise, we utilized a simple first-order differential equation:$$\sin (x+y) \frac{dy}{dx} = 5$$

In such an equation, solving usually involves isolating the function and its derivative and then integrating both sides. This exercise cleverly incorporates the concept of differentials, where we have the relationship $$dt = dx + dy$$
. By comparing the given differential equation with the integral we have, we integrate to find a function that describes the relationship between $$x$$and $$y$$.

Understanding differential equations is essential, as it provides a powerful toolset for solving a plethora of real-world problems, where the rate of change (represented by derivatives) is linked to the value of some function of interest.