Question

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

Short answer

\(x + C\)

Step-by-step solution

Understand the Integral of a Constant Function

The integral of a constant function, where the constant is 1 with respect to x, is equal to the variable x plus a constant C, since the derivative of x with respect to x is 1.

Integrate the Function

To find the indefinite integral \( \int dx \), we integrate 1 with respect to x which is x. We add a constant term C to represent the constant of integration.

Key concepts

Integral Calculus

At its core, integral calculus is a branch of mathematics focused on finding the sizes of things. It allows us to calculate areas, volumes, and sums that might be too complex or impossible to determine using basic algebra or geometry. Imagine you have a curve representing a car's speed over time, and you want to find out how far the car traveled in total. Integral calculus is the tool you'd use to figure that out.

It's composed of two main operations: definite and indefinite integrals. A definite integral gives you the total accumulation between two points, like the exact distance traveled by the car over a certain time frame. An indefinite integral, on the other hand, is more like a formula that gives you all possible totals; it's represented by the antiderivative and includes a constant of integration.

Constant of Integration

Now, let's dive into the constant of integration. When you work out an indefinite integral, you're essentially undoing a derivative. Since derivatives of constant terms vanish, when you integrate, you need to account for that lost information.

That's where the constant of integration comes in. It's denoted as 'C' and represents all the possible constant values that could have been present in the function before differentiation. For example, if you differentiated x + 5 or x - 3, you'd get the same result: 1. When integrating back, how do you know which constant was originally there? You technically don't, which is why you add 'C' to cover all your bases and represent those infinite possibilities.

Antiderivative

The concept of an antiderivative is similar to doing reverse engineering on functions. If differentiation is about breaking a function down to find its rate of change, then taking an antiderivative is putting those pieces back together to rebuild the original function.

In simple terms, an antiderivative of a function is another function whose derivative is the original function. For the exercise, \( \int dx \), the antiderivative is 'x.' It's like asking, what function, when differentiated, will give us 1? The answer is 'x' because the slope of the line y=x is constant at 1. Remembering this helps you grasp why the antiderivative of \( \int dx \) is simply 'x.'

Integration Techniques

When integrating more complex functions, integration techniques become your mathematical toolbox. They're the methods you use to find antiderivatives when the function isn’t a basic polynomial, exponential, or trigonometric function.

This includes methods like substitution, which works like a puzzle, finding parts of an integral you can replace with a simpler variable to make the integration easier. Then there's integration by parts, which is like the reverse of the product rule in differentiation. There's also partial fraction decomposition for rational functions, and trigonometric substitution for integrals involving the square roots of quadratic expressions.

Each technique is a strategy to simplify complex integrals into forms that are easier to manage, allowing you to find the antiderivative that might not be immediately obvious.