Problem 63 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \sin (2 x) d x $$

Step-by-Step Solution

Answer

\( \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \)

Step 1: Identify the integral form

Recognize that the integral \( \int \sin(2x) \, dx \) involves a trigonometric function. We can use the integral formula \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \), where \( C \) is the constant of integration.

Step 2: Apply the formula

Substitute \( a = 2 \) into the formula \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \). This gives us \( \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \).

Step 3: Write the final solution

Conclude that the indefinite integral of \( \sin(2x) \) is \( -\frac{1}{2} \cos(2x) + C \). Here, \( C \) represents the constant of integration, which accounts for the family of antiderivatives.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve integrating expressions that are composed of trigonometric functions such as sine, cosine, tangent, and their multiples. These types of integrals are common in calculus and frequently appear in mathematical problems involving periodic phenomena like waves, oscillations, and circular motion. When dealing with

• sine and cosine functions,
• especially when their argument is a multiple of a variable,

we utilize specific integration formulas. Such formulas simplify and expedite the process. For instance, if you encounter an integral involving

• \(\int \sin(ax) \, dx\) which evaluates to \(-\frac{1}{a} \cos(ax) + C\)

this is due to the fact that differentiation of the cosine function

• gives back a negative sine function, and
• the coefficient \(a\) accounts for the chain rule during derivative operations.

Thus, understanding these foundational formulas accelerates solving trigonometric integrals.

Integration Techniques

Integration techniques are various methods utilized to find antiderivatives or integrals of functions that are not immediately obvious. While some functions have straightforward integrals, others require more strategic approaches. One critical technique involves identifying the form of the integrand.In our example

• \( \int \sin(2x) \, dx\),

we apply the simple substitution technique. Here, we recognize the form \( \int \sin(ax) \, dx\) and leverage a known formula to solve the integral.Applying substitution can sometimes be necessary when the direct application of a formula is not apparent. Variables are substituted in such a way that the resulting integral is easier to solve, and then the substitution is reversed to get the result back in terms of the original variable. Techniques such as substitution, integration by parts, and trigonometric identities are crucial tools in solving integrals efficiently.

Constant of Integration

The constant of integration, denoted by \(C\), is a crucial component when solving indefinite integrals. Whenever you perform integration without specific boundaries, i.e., indefinite integrals, you include this constant.Why include it? While differentiation is precise, integration can yield infinitely many solutions. Consider:

• The integral \(\int f(x) \, dx\) can lead to a whole family of functions, differing only by a constant.
• For example, \(\int \sin(2x)\, dx = -\frac{1}{2} \cos(2x) + C\) respects this family by ensuring that any possible shift along the y-axis is accounted for.

Thus, the constant of integration reminds us that multiple antiderivatives exist for any indefinite integral and guarantees completeness of the solution.