Question

Find each indefinite integral. Check some by calculator. $$\int x \, d x$$

Short answer

\[ \int x \, dx = \frac{x^2}{2} + C \]

Step-by-step solution

Apply the Power Rule for Integration

The power rule for integration states that the integral of x raised to the n, written as \( x^n \), is equal to \( \frac{x^{n+1}}{n+1} \) plus a constant of integration C, provided that n is not equal to -1. In this case, the integral is \( \int x \, dx \) which implies n is 1.

Integrate the Function

Applying the power rule from step 1, we increase the exponent by 1 and divide by this new exponent: \( \int x \, dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C \).

Write the Final Answer

The final expression for the indefinite integral is \( \frac{x^2}{2} + C \) where C represents the constant of integration.

Key concepts

Power Rule for Integration

Understanding the power rule for integration is key to solving many indefinite integrals, particularly those that involve polynomials. The power rule is a straightforward technique that applies to integrals of the form \( \int x^n \, dx \), where \( n \) is a real number other than -1. This is because integrating \( \frac{1}{x} \) yields a logarithmic function, not a power function.

To apply the power rule for integration, simply increase the exponent by one and then divide by this new exponent. That is, \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. This technique is derived from the reverse process of differentiation, where the power rule is also used but in the opposite manner. For example, if you integrate \( x \), which is \( x^1 \), according to the power rule, you elevate the exponent by 1, resulting in \( x^2 \), and then divide by 2, giving you \( \frac{x^2}{2} \). Remember to always include the constant of integration to account for any constants that disappear during differentiation.

Constant of Integration

The constant of integration, commonly represented as \( C \), is an essential part of solving indefinite integrals. When we find the indefinite integral of a function, we are essentially reversing the differentiation process. However, since differentiation removes any constant terms (since their derivative is zero), we must consider that there could have been a constant present in the original function.

That's where the constant of integration comes into play. Each time we integrate, we add \( C \) to the result to indicate that there might be any number of constants that could have been in the original function. It represents all the possible vertical shifts of the antiderivative. When evaluating definite integrals, the constant of integration does not play a role since it cancels out, but it is crucial for indefinite integrals as it represents an infinite family of antiderivatives.

Integration Techniques

While the power rule is an effective method for integrating functions of the form \( x^n \), there are a variety of other integration techniques that are useful for different types of functions. These include substitution (often helpful when dealing with function compositions or trigonometric identities), integration by parts (useful for products of functions), and partial fractions (which simplifies rational functions into simpler fractions).

Besides these, there are also special techniques for trigonometric integrals, trigonometric substitutions, and improper integrals. Each technique has its own set of rules and can be selectively applied depending on the function being integrated. Advanced calculus often requires a combination of these methods to tackle more complex integrals. It is important for students to practice these techniques and learn when and how to apply them effectively to solve indefinite integrals.