Question

In Exercises \(1-6,\) find the indefinite integral. $$\int x^{-2} d x$$

Short answer

The indefinite integral of \(x^{-2}\) dx is \(\frac{1}{x} + C\).

Step-by-step solution

Identify the type of integral

The integral is \(\int x^{-2} dx\). Here, -2 is the exponent on the variable x, which means this integral is a power rule problem when n ≠ -1.

Apply the power rule for integration

To apply the power rule, we add 1 to the exponent and divide by this new number. The antiderivative of \(x^{-2}\) is \(-\frac{x^{(-2+1)}}{-2+1}\) which simplifies to \(-\frac{x^{-1}}{-1}\) or simply \(\frac{1}{x}\).

Add the constant of integration

As we are finding indefinite integral, need to add a constant, typically represented as 'C'. So, our final answer is \(\frac{1}{x} + C\).

Key concepts

Power Rule for Integration

Understanding the power rule for integration is essential for dealing with indefinite integrals where the integrand is a power of x. When you're faced with an integral such as \(\int x^n dx\), with n not equal to -1, you apply the power rule by increasing the exponent by one and then dividing by this new exponent. The general form looks like: \[\int x^n dx = \frac{x^{n+1}}{n+1} + C\], where C represents the constant of integration.

In the example \(\int x^{-2} dx\), we actually use the power rule backwards because the exponent is -2. Adding one to the exponent gives us -1, and we divide by this new number, resulting in \(-\frac{1}{x}\). The power rule simplifies the process and allows quick computation of indefinite integrals, saving time and reducing complexity in calculus problems.

Antiderivative

An antiderivative of a function f(x) is a function F(x) whose derivative is f(x). In other words, antiderivatives are the reverse process of taking derivatives. When you find the antiderivative of a function, you are performing the operation of integration. For instance, if you are given the function f(x) = x^n, the antiderivative F(x), using the power rule, would be \(\frac{x^{n+1}}{n+1}\) plus the constant of integration.

This concept is fundamental because it ties together the area under a curve with an original function, providing a bridge between differential and integral calculus. The process of finding antiderivatives is not always straightforward, and it requires familiarity with a variety of integration techniques, including the power rule, substitution, parts, and others.

Constant of Integration

The constant of integration is a critical aspect of indefinite integrals. Whenever we find an antiderivative of a function, it's important to include the constant of integration 'C'. This is because the derivative of a constant is zero, and thus the antiderivative is not unique.

For example, if you take the derivative of the functions F(x) = \(\frac{1}{x} + 3\) and G(x) = \(\frac{1}{x} - 5\), you get the same result: f(x) = -\(\frac{1}{x^2}\). Both F(x) and G(x) are antiderivatives of f(x), and they differ only by a constant. When we integrate to get the antiderivative, we must account for all possibilities by adding the plus C at the end of the expression, such as in the integral \(\int x^{-2} dx = -\frac{1}{x} + C\). Not doing so would imply there's only one antiderivative, which is simply not true.