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6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation

Syllabus
2020
Topic
6.8
Level
—

Learning objectives

—FUN-6.C—Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives• FUN-6.C Determine antiderivatives of functions and indefinite integrals, using knowledge of derivatives.• FUN-6.C.1 ∫ f(x)dx is an indefinite integral of the function f and can be expressed as ∫ f(x)dx = F(x) + C, where F′(x) = f(x) and C is any constant.• FUN-6.C.2 Differentiation rules provide the foundation for finding antiderivatives.• FUN-6.C.3 Many functions do not have closed-form antiderivatives.• Enduring understanding FUN-6: Recognizing opportunities to apply knowledge of geometry and mathematical rules can simplify integration.

Reverse Derivative Rules to Find Antiderivatives

The indefinite integral ∫f(x) dx\int f(x)\,dx means the complete family F(x)+CF(x)+C whose derivative is f(x)f(x). The constant is necessary because all vertical shifts of FF have the same derivative.

Integrand Antiderivative
xnx^n, n≠−1n\ne-1 xn+1/(n+1)+Cx^{n+1}/(n+1)+C
1/x1/x, x≠0x\ne0 ln⁡∣x∣+C\ln|x|+C
exe^x ex+Ce^x+C
cos⁡x\cos x sin⁡x+C\sin x+C
sin⁡x\sin x −cos⁡x+C-\cos x+C

Reverse the matching derivative rule for each term. Constants factor out and sums integrate term by term. After combining the terms, write one +C+C for the entire antiderivative family and differentiate the result to verify it.

For x≠0x\ne0, ∫(6x2−4/x+3ex) dx=2x3−4ln⁡∣x∣+3ex+C\int(6x^2-4/x+3e^x)\,dx=2x^3-4\ln|x|+3e^x+C. Differentiating gives 6x2−4/x+3ex6x^2-4/x+3e^x, exactly the original integrand, so the coefficients, signs, and logarithm condition are consistent.

Not every function has a closed-form antiderivative; for example, e−x2e^{-x^2} cannot be expressed using the usual finite collection of elementary functions. That does not mean a related definite integral is meaningless: it may still be represented by an accumulation function or approximated numerically.