Antiderivatives and Indefinite Integrals

In calculus, finding an antiderivative is the reverse process of finding a derivative. If you know the rate at which a quantity is changing, the antiderivative allows you to reconstruct the original quantity.

What is an Antiderivative?

A function $F(x)$ is considered an antiderivative of $f(x)$ if taking the derivative of $F(x)$ gives you back $f(x)$: $F'(x) = f(x)$

For example, if $f(x) = 2x$, an antiderivative is $F(x) = x^2$ because the derivative of $x^2$ is exactly $2x$.

The Constant of Integration (+ C)

Notice that the derivative of $x^2 + 5$ is also $2x$. The same is true for $x^2 - 10$ or $x^2 + 42$. Because the derivative of any constant is zero, there are infinitely many antiderivatives for any given function.

To represent the entire family of possible antiderivatives, we use an indefinite integral and add an arbitrary constant $C$, known as the constant of integration: $\int f(x) \, dx = F(x) + C$

Here, $\int$ is the integral sign, $f(x)$ is the integrand (the function being integrated), and $dx$ indicates the variable of integration.

Basic Integration Rules

Here are a few essential rules for evaluating indefinite integrals:

• Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)
• Exponential: $\int e^x \, dx = e^x + C$
• Reciprocal: $\int \frac{1}{x} \, dx = \ln|x| + C$
• Sine: $\int \sin x \, dx = -\cos x + C$
• Cosine: $\int \cos x \, dx = \sin x + C$
• Sum and Difference: $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$

Example Problems

Example 1: Find $\int (3x^2 - 4x + 2) \, dx$

Apply the sum/difference rule and the power rule to each term individually: $\int 3x^2 \, dx = 3 \left(\frac{x^3}{3}\right) = x^3$ $\int 4x \, dx = 4 \left(\frac{x^2}{2}\right) = 2x^2$ $\int 2 \, dx = 2x$

Combine the terms and don't forget to add the constant of integration: $\int (3x^2 - 4x + 2) \, dx = x^3 - 2x^2 + 2x + C$

Example 2: Find $\int (e^x + 2\sin x - \frac{1}{x}) \, dx$

Integrate each term using the basic rules: $\int e^x \, dx = e^x$ $\int 2\sin x \, dx = -2\cos x$ $\int \frac{1}{x} \, dx = \ln|x|$

Combine the results to find the final indefinite integral: $\int \left(e^x + 2\sin x - \frac{1}{x}\right) \, dx = e^x - 2\cos x - \ln|x| + C$