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Calculus Integration

Integration rules

20 practice questions 0 video lessons Theory + worked examples

Integration Rules

California Calculus • Standard 17.0 • Integration

Integration Rules is a topic in Integration in the California Calculus Standards. It is aligned to Standard 17.0, which requires students to compute, by hand, the integrals of a wide variety of functions.

The integration rules — the reverse power rule, linearity, and the standard integrals of \(\dfrac{1}{x}\), \(e^x\), \(\sin x\), and \(\cos x\) — antidifferentiate a wide range of functions by hand.

California Calculus › Integration › Integration Rules  —  Standard 17.0

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Theory

The integration rules let you antidifferentiate a wide range of functions by hand. The reverse power rule plus linearity (constant-multiple and sum) handles polynomials and roots, and a short list of standard integrals covers \(\dfrac{1}{x}\), \(e^{x}\), \(\sin x\), and \(\cos x\).

Integration is linear, so it splits over sums and pulls out constants:

\[\int \big(a\,f+b\,g\big)\,dx=a\int f\,dx+b\int g\,dx.\]

The reverse power rule does most of the work; rewrite roots and reciprocals as powers so it applies.

A few standard integrals must simply be known: \(\dfrac{1}{x}\to\ln|x|\), \(e^{x}\to e^{x}\), \(\cos x\to\sin x\), \(\sin x\to-\cos x\).

Key idea: you cannot integrate a product or quotient term by term. Simplify first (expand, or split the fraction) so linearity applies.
The area under a curve found with the integration rules The power, constant multiple, and sum rules integrate a polynomial term by term to give the area under it. x y area
The rules give the area under a polynomial.
The reciprocal curve whose integral is a logarithm The integral of one over x is the natural logarithm, the one case the power rule cannot handle. x y y = 1/x
\(\displaystyle\int \dfrac{1}{x}\,dx=\ln|x|+C\) — the power-rule exception.

The core rules:

\[\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\ (n\neq -1),\qquad\int \dfrac{1}{x}\,dx=\ln|x|+C\]
reverse power rule and the integral of one over x
\[\int e^{x}\,dx=e^{x}+C,\quad\int \cos x\,dx=\sin x+C,\quad\int \sin x\,dx=-\cos x+C\]
standard integrals of e to the x, cosine, and sine
Watch the sign: \(\int\sin x\,dx=-\cos x+C\) — the minus sign is easy to drop.

How to integrate with the basic rules

  1. Simplify first: expand products and split quotients into separate terms.
  2. Rewrite roots and reciprocals as powers.
  3. Integrate term by term, using the power rule and standard forms, then add \(+C\).
Example 1 — Term by term
Find \(\displaystyle\int (3x^2-4x+5)\,dx\).
Solution

Integrate each term with the power rule (the constant \(5\) becomes \(5x\)).

\(\int (3x^2-4x+5)\,dx\)\(=\)\(3\cdot\dfrac{x^3}{3}-4\cdot\dfrac{x^2}{2}+5x+C\)
\(=\)\(x^3-2x^2+5x+C\)
integral is x cubed minus 2 x squared plus 5x plus C
Example 2 — A root
Find \(\displaystyle\int \dfrac{1}{\sqrt{x}}\,dx\).
Solution

Write \(\dfrac{1}{\sqrt{x}}=x^{-1/2}\). Add one to the exponent (\(-\dfrac{1}{2}+1=\dfrac{1}{2}\)) and divide.

\(\int x^{-1/2}\,dx\)\(=\)\(\dfrac{x^{1/2}}{1/2}+C\)
\(=\)\(2x^{1/2}+C=2\sqrt{x}+C\)
integral equals two root x plus C
Example 3 — Split a quotient
Find \(\displaystyle\int \dfrac{x^2+1}{x}\,dx\).
Solution

Divide each term by \(x\) first.

\(\dfrac{x^2+1}{x}\)\(=\)\(x+\dfrac{1}{x}\)
\(\int\)\(=\)\(\dfrac{x^2}{2}+\ln|x|+C\)
integral is x squared over 2 plus natural log of x plus C
Example 4 — Standard forms
Find \(\displaystyle\int (e^{x}+\cos x)\,dx\).
Solution

Use the standard antiderivatives.

\(\int (e^{x}+\cos x)\,dx\)\(=\)\(e^{x}+\sin x+C\)
integral is e to the x plus sin x plus C

Common pitfalls

You cannot integrate a quotient directly. Split \(\dfrac{x^2+1}{x}\) into \(x+\dfrac{1}{x}\) first.
The \(n=-1\) case is special. \(\int\dfrac{1}{x}\,dx=\ln|x|+C\), not a power.
Sign of \(\int\sin x\). It is \(-\cos x+C\); losing the minus sign is a common error.

Frequently asked questions

What is the reverse power rule for integration?

\(\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\) for \(n\neq -1\): add one to the exponent and divide by the new exponent.

What is the integral of 1/x?

\(\int\dfrac{1}{x}\,dx=\ln|x|+C\). This is the one power the reverse power rule cannot handle.

How do you integrate a fraction like (x^2+1)/x?

Split it into separate terms first: \(x+\dfrac{1}{x}\), then integrate each to get \(\dfrac{x^2}{2}+\ln|x|+C\).

What are the integrals of sin x and cos x?

\(\int\cos x\,dx=\sin x+C\) and \(\int\sin x\,dx=-\cos x+C\). Note the minus sign on the sine integral.

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