Question

Perform the indicated integrations. $$ \int \frac{\sin \sqrt{t}}{\sqrt{t}} d t $$

Short answer

The integral is \(-2 \cos \sqrt{t} + C\).

Step-by-step solution

Recognize the Integration Technique

First, observe the integral: \( \int \frac{\sin \sqrt{t}}{\sqrt{t}} \, dt \). The structure resembles a known integration form that often uses a substitution to simplify.

Make a Substitution

Use the substitution \( u = \sqrt{t} \). Then, \( du = \frac{1}{2\sqrt{t}} \, dt \) or \( dt = 2u \, du \). Replace the variables in the integral: the original integral \( \int \frac{\sin \sqrt{t}}{\sqrt{t}} \, dt \) becomes \( \int 2 \sin u \, du \) after substitution.

Integrate the Simplified Expression

The simplified integral is \( 2 \int \sin u \, du \). Use the basic trigonometric integration formula: \( \int \sin u \, du = -\cos u \). Thus, \( 2 \int \sin u \, du = -2 \cos u + C \), where \( C \) is the constant of integration.

Substitute Back to Original Variable

Now, replace \( u \) back with \( \sqrt{t} \): \( -2 \cos u + C \) becomes \( -2 \cos \sqrt{t} + C \). This expression represents the original integral's antiderivative.

Key concepts

Integration Techniques

Integration is a fundamental concept in calculus used to find areas, volumes, and other quantities. It involves calculating the accumulation of quantities, which makes it incredibly useful in many fields. There are several methods or techniques of integration that help solve different types of integrals.
Some common integration techniques include:

• Substitution: This technique simplifies the integral by substituting parts of the integral with a single variable, making it easier to solve. It is particularly useful when dealing with composite functions.
• Integration by Parts: Based on the product rule of differentiation, this method is especially helpful for integrals that are products of functions, like polynomial and exponential functions.
• Partial Fraction Decomposition: Useful for rational functions, where the numerator and denominator are polynomials. It breaks down complex fractions into simpler parts.

Choosing the right technique is crucial for solving integrals effectively. Observing the integral's structure usually gives clues about which method to apply.

Trigonometric Integration

Trigonometric integration is a powerful skill when dealing with integrals involving trigonometric functions. Understanding trigonometric identities and how to manipulate them is key to simplifying and solving these integrals.
Common strategies to tackle trigonometric integrals include:

• Using trigonometric identities to simplify the integral before integrating. For example, identities like \(\sin^2 x + \cos^2 x = 1\) can transform the integral into a simpler form.
• Substitution can also be beneficial, especially when the argument of the trigonometric function is complex, like \(\sin(\sqrt{t})\).
• Familiarize yourself with integrals of basic trigonometric functions, such as \(\int \sin x \, dx = -\cos x + C\) or \(\int \cos x \, dx = \sin x + C\).

It's helpful to practice recognizing patterns and employing various trigonometric identities to aid in solving integrals. The more familiar you are with these patterns, the easier it becomes to integrate trigonometric functions.

Substitution Method

The substitution method is a crucial technique in integral calculus, often used to simplify integrals by changing the variable of integration. It works by introducing a new variable to transform the integral into an easier form to integrate.
Here’s how it generally works:

• Identify a part of the integral that can be replaced with a single variable. This usually involves the inner function of a composition, which allows us to use trigonometric simplifications or simplify a fraction.
• Derive the differential of that new variable to aid in substitution. This helps in adjusting the bounds or the differential of the integral.
• Substitute back the original variable once the integration of the substitute variable is complete.

In the example integral, \(\int \frac{\sin \sqrt{t}}{\sqrt{t}} \, dt\), we use substitution to simplify the function by letting \(u = \sqrt{t}\). This transforms the original integral into \(2\int \sin u \ du\), which is a basic trigonometric integral. Understanding when and how to employ substitution can significantly reduce the complexity of an integral.