Chapter 5

# There Must Be a Better Way — Introducing the Indefinite Integral

IN THIS CHAPTER

Seeing how the indefinite integral is the inverse of the derivative

Clarifying the differences between definite and indefinite integrals

In Chapter 4, I discuss the Riemann sum formula, which provides the formal definition for the definite integral. Although this formula can be used to calculate the definite integral, it usually results in lengthy and difficult calculations.

There must be a better way! And, indeed, there is.

In this chapter, I introduce you to the Fundamental Theorem of Calculus (FTC), which provides the link between the slope of a curve (the derivative) and the area under it (the integral). This connection provides a way to calculate definite integrals without resorting to the Riemann sum formula. Instead, you use the FTC to evaluate integrals as antiderivatives — that is, by understanding integration as the inverse of differentiation.

This insight leads an important new concept: the indefinite integral. The indefinite integral looks similar to the definite integral but provides the power to calculate the values of infinitely many related definite integrals using anti-differentiation.

Spoiler alert: Using indefinite integrals is the most common way you’ll evaluate definite ...

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