Question

In Exercises \(41-44,\) find or evaluate the integral using substitution first, then using integration by parts. $$ \int \sin \sqrt{x} d x $$

Short answer

The result of the integral \( \int \sin \sqrt{x} d x \) by substitution method is \( -2x \cos(\sqrt{x}) + 2 \sin(\sqrt{x}) + C\). Attempting to solve this integral by integration by parts does not simplify the integral, rather it makes the integral more complex. Therefore, substitution method is more suitable for this problem.

Step-by-step solution

Evaluate by Substitution

The substitution method is considered first. Let \(u = \sqrt{x}\),then \(x = u^2\) and \(dx = 2u du\). The integral can then be rewritten as:\[\int 2u \sin u du\]Now we can integrate using a standard rule for the integral of \(u \sin u\), which is \(-u \cos u + \sin u + C\). Replace the \(u = \sqrt{x}\) we get:\[-2x \cos(\sqrt{x}) + 2 \sin(\sqrt{x}) + C \]

Evaluate by Integration By Parts

Then, consider the integration by parts method. In this, we need to choose 'u' and 'dv' such that the resulting integral is simpler than the original. This can sometimes involve some creative problem solving. Let u = \(\sin \sqrt{x}\), then \(du = \frac{\cos \sqrt{x}}{2\sqrt{x}} dx\). Let dv = \(dx\), then \(v = x\). Integration by parts formula gives us \(uv - \int v du\), so the integral becomes:\[x \sin \sqrt{x} - \int x * \frac{\cos \sqrt{x}}{2\sqrt{x}} dx \]Unfortunately, this does not simplify the integral, and instead appears to introduce an additional complexity. In general, if the integral becomes more complex when attempting integration by parts, it is an indication that another method (such as substitution) would be more suitable.

Key concepts

Integration Techniques

Integration techniques are various methods used to calculate the integral of functions that may not be straightforward. The role of integration techniques is much like having a toolbox – each tool serves a unique purpose and is employed depending on the specific nature of the problem at hand.

Among the many techniques available, some of the most commonly taught in calculus courses are u-substitution and integration by parts. Both are powerful methods that can be used to break down complex integrals into simpler, more manageable components.

Choosing the right technique often requires recognizing patterns in the integrand and may involve trial and error. In the context of an educational platform, it's crucial to encourage students not only to master each technique but also to develop intuition on when each technique is most effectively applied.

U-Substitution

U-substitution, often referred to as a change of variable, is a method for finding indefinite integrals that resemble the chain rule in reverse. The basic idea is to simplify an integral by substituting a part of the integrand with a new variable, commonly denoted as 'u'.

This substitution makes the integral easier to evaluate, typically transforming it into one of the standard integrals that students are familiar with. The process involves three steps:

• Choose a substitution that simplifies the integral.
• Express the differential dx in terms of du.
• Rewrite the integral in terms of u and integrate.

The success of u-substitution hinges on the student's ability to identify a part of the integrand that, when derived, appears elsewhere in the integrand. This identification often simplifies the integral significantly, leading to an easier path to the solution.

Integration by Parts

Integration by parts is a technique that stems from the product rule of differentiation. The fundamental idea behind it is to transform the integral of a product of two functions into an integral that is, hopefully, simpler. The formula for integration by parts is given by \( uv - \int v \, du \), where one function is differentiated (u becomes du) and the other is integrated (dv becomes v).

To effectively apply integration by parts, one must select u and dv wisely. A common mnemonic used to remember which function to choose as 'u' is LIATE: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions, listed in order of preference.

While this technique can be incredibly useful, it may sometimes lead to an integral more complex than the original. When this occurs, it's advisable to explore other techniques, such as u-substitution, which might offer a more straightforward path to the solution.

Indefinite Integral

The indefinite integral, represented by the integral symbol without upper and lower limits, is a fundamental concept in calculus that corresponds to finding the antiderivative of a function. It represents the collection of all antiderivatives of a function and is essential in calculating the area under a curve. The process of finding an indefinite integral is known as integration.

Indefinite integrals are expressed with a constant of integration, denoted by 'C', which acknowledges that any constant added to a function's derivative results in the same original function post-derivation. The indefinite integral's role in problems often relates to reversing the process of differentiation to find the original function, given its derivative. In learning platforms, presenting a variety of problems that utilize indefinite integrals in their solutions can greatly enhance student understanding and mastery of integration techniques.