Recommended resources
The freely available resources this lesson will direct you to.
Hello! This first module is about finding antiderivatives: functions whose derivatives you already know. We will begin with expressions that become manageable through algebra—expanding, separating terms, or rewriting roots and fractions. Later lessons will introduce techniques for expressions that cannot be handled by those changes alone.
By the end of this lesson, you should be able to turn an integrand into familiar forms, integrate those forms, and check your answer by differentiating it.
What an indefinite integral asks for
If , then is an antiderivative of . An indefinite integral asks for all such functions:
Here is the integrand, is the variable of integration, and is an arbitrary constant. For example, both and have derivative , so
The constant is not an optional finishing touch: it records the whole family of answers.
Read the following sections to connect this notation to differentiation. In the formula table, concentrate on the forms used below rather than trying to memorize every row at once.
4.1: Antiderivatives and Indefinite Integrals
Read “4.1: Antiderivatives and Indefinite Integrals” from LibreTexts for the meaning of an antiderivative, the constant of integration, and a reference table of basic forms.
In “The Reverse of Differentiation,” read the opening explanation, then the “General Form of an Antiderivative” theorem immediately after it; stop before “Finding Antiderivatives.” In “Indefinite Integrals,” read from the notation through the paragraph describing the family for 2x; stop before the figure. Finally, read the power rule and consult “Table 1: Integration Formulas.” Focus on the rows for constants, powers, 1/x, e^x, sine, cosine, \sec^2x, and the two basic inverse-trigonometric forms. You can leave the hyperbolic-function rows for another time.
The main power rule comes directly from reversing the derivative of a power:
For , its denominator would be zero. That case has its own form:
The absolute value matters when is negative. An antiderivative is considered on an interval where the integrand is defined; for , such an interval cannot cross zero.
Here is a compact set of other forms to recognize. Each can be checked by differentiating the expression on the right.
| Integrand | Antiderivative, before adding |
|---|---|
| , a constant | |
| , where |
The task is not always to spot one of these forms immediately. Often you must first reveal it.
Why rewriting is allowed
Differentiation distributes over sums and respects constant multiples. Integration therefore has the corresponding properties: if and , then
for constants and . This lets you integrate a polynomial term by term. It does not say that you can integrate the factors of a product separately, or integrate the numerator and denominator of a quotient separately.
4.1: Antiderivatives and Indefinite Integrals
Return to the LibreTexts reading for the reason sums and constant multiples can be handled term by term.
In “Indefinite Integrals,” begin just after the checkpoint about x\cos x. Read the properties, including the “Sums and Differences” and “Constant Multiples” statements. Stop before “Evaluating Indefinite Integrals.” Notice that the justification comes from differentiating a proposed answer.
Consider a radical alongside an exponential:
The first rewrite exposes two powers. Apply the power rule to each, then use the exponential form:
Notice the exponent in the second term. It is eligible for the power rule; only is the exception.
Expand or divide before matching forms
A product of polynomials may not look like a sum of standard forms, but expanding it makes one. Likewise, a fraction with a single monomial in its denominator can often be split term by term across its numerator.
Watch the worked examples with that decision in mind: What algebraic move makes the integrand recognizable?
Watch “Indefinite Integral” from The Organic Chemistry Tutor for worked examples of the power rule, radicals, polynomial products, and fractions with monomial denominators.
Watch the power rule and radical rewriting. Then watch polynomial expansion and monomial division. Focus on the step before integration in each example. In the final example, the video writes \ln x; use \ln|x| for a general antiderivative on intervals where x\ne0.
For instance, expand this product rather than looking for an “integral of a product” rule:
The same principle works for a quotient when each numerator term can be divided by the entire monomial denominator:
Now the middle term calls for the logarithm, while the other two use the power rule:
Be careful about what made this split valid: the denominator was the single term . A denominator such as cannot be distributed across numerator terms in this way.
Checking an answer—and recognizing the boundary
Differentiation is a reliable final check. In the quotient example, differentiating the proposed answer gives
which is exactly the rewritten integrand. This check catches missed coefficients, signs, and powers.
A useful routine for this lesson is to simplify the integrand algebraically, identify each standard form, integrate term by term where permitted, and differentiate the result. If straightforward rewriting still leaves a composite expression—for example, a familiar function applied to a more complicated inner expression—do not force the power rule onto it. The next lesson develops substitution for that situation.
You can now evaluate many indefinite integrals by making their structure visible first. Remember the power rule’s exception, include , and use differentiation to verify the result. Next, you will learn how to handle integrands whose inner expression and its derivative are the key to finding an antiderivative.