Question

In Exercises \(1-10,\) find the indefinite integral. $$\int x \sin x d x$$

Short answer

\(-x \cos x + \sin x + C\)

Step-by-step solution

Identifying the Terms

Identify the terms in the integral that will form \(u\) and \(dv\). In \(\int x \sin x dx\), take \(u = x\) and \(dv = \sin x dx\). The choice is guided by the LIATE rule which prefers algebraic functions like \(x\) for \(u\).

Finding Du and V

This step is about finding the differential \(du\) and the function \(v\). Differentiate \(u\) to get \(du\), and integrate \(dv\) to find \(v\).We find \(du = dx\) and \(v = -\cos x\).

Apply Integration by Parts

Substitute \(u, v, du,\) and \(dv\) into the integration by parts formula \(\int u dv = uv - \int v du\). The result is \[ \int x \sin x dx = x(-\cos x) - \int -\cos x dx \] Simplifying gives \[ - x \cos x + \int \cos x dx \]

Integrate the Remaining Term

The second term on the right side of the equation, \(\int \cos x dx\), is a basic integral. Its result is \(\sin x\). Now replace the integral with its result: \[ - x \cos x + \sin x \] This is the indefinite integral. Because with indefinite integrals, there could be a constant \(C\) added (a result of the constant of integration), the full solution can be written as \[ - x \cos x + \sin x + C \]

Key concepts

Integration by Parts

Integration by parts is a crucial technique in integration, a part of differential calculus, that is used for finding the integral of the product of two functions. This method is based on the product rule for differentiation and provides a way to simplify integrals by breaking them down into more manageable parts.

The general formula for integration by parts is \[ \int u dv = uv - \int v du \], where \(u\) and \(dv\) are parts of the original integral you choose to designate. The idea is to differentiate \(u\) to get \(du\) and integrate \(dv\) to get \(v\), and then apply the formula. In practice, you should select \(u\) and \(dv\) such that the resulting integral \(\int v du\) is simpler than the original. A good exercise improvement is to practice this selection process, as it is often where students struggle. Remember, with more practice comes a better intuition for the choices that simplify an integral.

LIATE Rule

The LIATE rule is an informal guideline that helps you decide how to choose \(u\) and \(dv\) for integration by parts. The rule is an acronym where LIATE stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions respectively.

The LIATE rule suggests a priority order for choosing the \(u\) function, with Logarithmic functions being the highest priority and Exponential the lowest. In our exercise example \( \int x \sin x dx \), \(x\) is an Algebraic function, and \(\sin x\) is a Trigonometric function. According to LIATE, we prefer \(x\) for \(u\) since Algebraic functions come before Trigonometric in the priority list. This way, after differentiation and integration, we aim for a simpler integral to solve.

Differential Calculus

Differential calculus is a subfield of calculus concerned with the study of how functions change when their inputs change. It's all about rates of change and slopes of curves. The main tool in differential calculus is the derivative, which measures how a quantity changes as another quantity changes.

In the context of our exercise, when integrating by parts, we use differential calculus to find \(du\), the derivative of \(u\). This process essentially reverses differentiation, taking us back to an integral form. Understanding how different functions differentiate is essential to mastering integration techniques and solving complex integral problems. Exercises in differential calculus will help sharpen your skills in computing derivatives accurately and effectively.

Antiderivative

An antiderivative of a function is another function that reverses the process of differentiation. In other words, if \(F'(x) = f(x)\), then \(F\) is an antiderivative of \(f\). Finding an antiderivative is the essence of integration. In integration problems, we often look for the antiderivative to determine the area under a curve or to solve differential equations.

Most functions have infinitely many antiderivatives, differing by a constant. Because of this, when we write the solution to an indefinite integral, such as our exercise \( -x \cos x + \sin x + C \), we include the constant of integration \(C\) to account for all possible antiderivatives. Understanding antiderivatives is key for tackling integration problems successfully, and practicing them can enhance your intuition for solving indefinite integrals.