Problem 22 - Calculus Single Variable

Question

Use the method of substitution to calculate the indefinite integrals. $$ \int \frac{\sin (t)-\cos (t)}{(\sin (t)+\cos (t))^{2}} d t $$

Step-by-Step Solution

Answer

The indefinite integral is \( \frac{1}{\sin(t) + \cos(t)} + C \).

Step 1: Choose a Substitution

Identify a substitution to simplify the integral. Notice that the denominator is in the form of \((\sin(t) + \cos(t))^2\). Let's make a substitution \(u = \sin(t) + \cos(t)\).

Step 2: Calculate the Derivative of the Substitution

Find \(du\) by differentiating \(u = \sin(t) + \cos(t)\) with respect to \(t\):\[ du = (\cos(t) - \sin(t)) \, dt \]

Step 3: Simplify the Integral with Substitution

Rewrite the integral using the substitution. Since \(du = (\cos(t) - \sin(t)) \, dt\), the integral becomes:\[ \int \frac{-1}{u^2} \, du \]Substitute \( -du \) for \((\sin(t) - \cos(t)) \, dt\).

Step 4: Integrate

Integrate \(\int \frac{-1}{u^2} \, du\):\[ \int \frac{-1}{u^2} \, du = \int -u^{-2} \, du \]Use the power rule for integration:\[ = u^{-1} + C = \frac{1}{u} + C \]

Step 5: Back Substitute to Original Variable

Replace \(u\) with the original expression \(\sin(t) + \cos(t)\):\[ \frac{1}{\sin(t) + \cos(t)} + C \]

Step 6: Present Final Solution

The integral \(\int \frac{\sin(t) - \cos(t)}{(\sin(t) + \cos(t))^2} \, dt\) is evaluated as:\[ \frac{1}{\sin(t) + \cos(t)} + C \]

Key Concepts

Indefinite Integrals

An indefinite integral, also known as an antiderivative, is a function that reverses the process of differentiation. It takes a function and finds another function whose derivative is the original function you started with. This resulting function is often represented with an integration symbol, \( \int \ \), along with a \( C \) that signifies the constant of integration.
When you're given the task of finding an indefinite integral, you're essentially required to determine the original function before it was differentiated. Since functions can have many different forms before differentiation, the constant \( C \) is essential in representing all of these possibilities.
When integrating, especially in complex cases, a variety of techniques may be used to simplify the process, such as substitution. This method can often transform the integral into a simpler form which is easier to evaluate.

Trigonometric Functions

Trigonometric functions, like \( \sin(t) \,\cos(t) \,\tan(t) \), play a critical role in calculus, especially as they often appear in integrals. These functions describe the relationships between the angles and sides of a triangle in a unit circle configuration.
When solving integrals that involve trigonometric expressions, like our given problem, understanding the basic identities and properties of these functions is crucial. For instance, knowing that the derivative of \( \sin(t) \) is \( \cos(t) \) and the derivative of \( \cos(t) \) is \(-\sin(t)\) provides insights when applying integration techniques.
In integrals, trigonometric functions might either be simplified using identities or transformed using substitution, which is a powerful approach in handling complex expressions.

Integration Techniques

There are numerous techniques to tackle various integrals, and substitution is one of the most powerful and commonly used methods. The idea is to simplify the integrand, which is the function to be integrated, by introducing a new variable. This can often change the form of the integral into something more manageable.
In our exercise, the substitution \( \u = \sin(t) + \cos(t) \) was chosen. This is because the expression appeared frequently in our integral, particularly in the denominator. This choice helped simplify the integral to a standard form that could be easily managed by using the power rule, \( \int u^{-2} \,du = -u^{-1} + C \).
Recognizing patterns and strategically choosing a substitution can significantly ease the process of integration, transforming complex problems into straightforward calculations. This serves as an efficient strategy for solving a wide range of integral problems.