Differentiation and Integration Formulas, CALC II – Study Notes
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Difficulty: Intermediate | Prerequisites: Algebra Foundations and Trigonometry notes.

Big Picture

Differentiation and integration formulas are the core toolkit of Calculus II. You learned many derivatives in Calc I; now you need the corresponding antiderivatives for integration. The integration formulas here are the building blocks that every technique in the course (substitution, integration by parts, partial fractions, trig substitution) depends on. If you cannot recall these from memory, every problem takes longer than it should.

TL;DR

There is a standard set of derivative formulas (power rule, exponentials, logarithms, all six trig functions, and the three inverse trig functions) paired with a matching set of basic integrals. Each integration formula is the reverse of a differentiation formula. Memorise the pairs and you can handle the straightforward integrals that appear inside more complex problems.

Key Terms

Derivative

The rate of change of a function at a point; geometrically, the slope of the tangent line. Written d/dx[f(x)] or f'(x).

Antiderivative

A function F(x) whose derivative is f(x). In simple terms, the reverse of differentiation. If F'(x) = f(x), then F is an antiderivative of f.

Indefinite integral

The general antiderivative of a function, written with the integral sign: the integral of f(x) dx = F(x) + C. Think of it as the family of all functions whose derivative is f(x).

Constant of integration (C)

The arbitrary constant added to every indefinite integral, because the derivative of any constant is zero. Forgetting +C is one of the most common mistakes on exams.

Core Content

Differentiation Formulas

  • d/dx [x^n] = n * x^(n-1)

  • d/dx [ln|x|] = 1/x

  • d/dx [e^x] = e^x

  • d/dx [a^x] = (ln a) * a^x

  • d/dx [sin(x)] = cos(x)

  • d/dx [cos(x)] = -sin(x)

  • d/dx [tan(x)] = sec^2(x)

  • d/dx [cot(x)] = -csc^2(x)

  • d/dx [sec(x)] = sec(x) tan(x)

  • d/dx [csc(x)] = -csc(x) cot(x)

  • d/dx [arcsin(x)] = 1 / sqrt(1 - x^2)

  • d/dx [arccos(x)] = -1 / sqrt(1 - x^2)

  • d/dx [arctan(x)] = 1 / (1 + x^2)

Basic Integration Formulas

  • Integral of x^n dx = x^(n+1) / (n+1) + C, where n is not equal to -1

  • Integral of (1/x) dx = ln|x| + C

  • Integral of e^x dx = e^x + C

  • Integral of a^x dx = a^x / ln(a) + C

  • Integral of sin(x) dx = -cos(x) + C

  • Integral of cos(x) dx = sin(x) + C

  • Integral of sec^2(x) dx = tan(x) + C

  • Integral of csc^2(x) dx = -cot(x) + C

  • Integral of sec(x) tan(x) dx = sec(x) + C

  • Integral of csc(x) cot(x) dx = -csc(x) + C

  • Integral of 1/(x^2 + 1) dx = arctan(x) + C

  • Integral of 1/sqrt(1 - x^2) dx = arcsin(x) + C

Useful Additional Integrals

  • Integral of tan(x) dx = ln|sec(x)| + C

  • Integral of sqrt(x) dx = (2/3) x^(3/2) + C

  • Integral of 1/(a^2 + u^2) du = (1/a) arctan(u/a) + C

Formulas

\frac{d}{dx}[x^n] = nx^{n-1}
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)
\frac{d}{dx}[\arcsin(x)] = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}[\arctan(x)] = \frac{1}{1+x^2}
\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C
\int \frac{1}{x^2+1} \, dx = \arctan(x) + C
\int \frac{1}{a^2+u^2} \, du = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C

Common Misconceptions

  • Forgetting the constant of integration (+C) on indefinite integrals. This costs marks on virtually every exam.

  • Confusing d/dx[tan(x)] = sec^2(x) with d/dx[sec(x)] = sec(x)tan(x). They look similar but are different formulas for different functions.

  • Applying the power rule to the integral of 1/x. The power rule gives x^0 / 0, which is undefined. The correct antiderivative is ln|x| + C.

  • Forgetting the negative sign on the derivatives of cos, cot, csc, and arccos. A useful pattern: the "co-" functions always pick up a negative.

Why It Matters / Exam Flags

  • You will use these formulas on nearly every exam question. They are the foundation for every integration technique in the course.

  • The inverse trig integrals (arcsin and arctan forms) frequently appear in trig substitution and partial fractions results.

  • The integral of tan(x) = ln|sec(x)| + C comes up often enough that it is worth memorising directly.

  • Watch for problems that look like power-rule integrals but have n = -1. That is the ln|x| case.

Quick Self-Test

  1. What is d/dx [sec(x)]?

  1. What is the integral of cos(x) dx?

  1. True or false: the integral of x^(-1) dx = x^0 / 0.

  1. What is d/dx [arctan(x)]?

  1. What is the integral of e^x dx?

Practice Q&A

Q: Find the integral of 3x^4 dx.

A: Apply the power rule: 3 * x^5 / 5 + C = (3/5)x^5 + C.

Q: Find d/dx [5 ln|x| + e^x - cos(x)].

A: Differentiate term by term: 5/x + e^x + sin(x).

Q: Evaluate the integral of sec^2(x) dx.

A: tan(x) + C.

Q: Find the integral of 1 / (1 + x^2) dx.

A: arctan(x) + C.

Q: Find the integral of 1 / sqrt(1 - x^2) dx.

A: arcsin(x) + C.

Connections to Other Topics

These formulas are the raw materials for every integration technique covered later in Calc II: u-substitution, integration by parts, trig substitution, and partial fractions all reduce a harder integral to one of these basic forms. The derivative formulas also return when working with Taylor and Maclaurin series, since building a series requires repeated differentiation.

Related Terms / Search Tags

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