Integration Formulas and the Net Change Theorem

Indefinite integral identities

Basic Properties of Indefinite Integrals

Constant Rule for indefinite integrals

If $c$ is a constant then $\int c\cdot f(x)dx=c\int f(x)dx$}}

Sum/Difference Rule for indefinite integrals

$\int {\Big (}f(x)+g(x){\Big )}dx=\int f(x)dx+\int g(x)dx$

$\int {\Big (}f(x)-g(x){\Big )}dx=\int f(x)dx-\int g(x)dx$}}

Indefinite integrals of Polynomials

Say we are given a function of the form, $f(x)=x^{n}$ , and would like to determine the antiderivative of $f$ . Considering that

${\frac {d}{dx}}{\frac {1}{n+1}}x^{n+1}=x^{n}$

we have the following rule for indefinite integrals:

Power rule for indefinite integrals

$\int x^{n}dx={\frac {x^{n+1}}{n+1}}+C$ for all $n\neq -1$

Integral of the Inverse function

To integrate $f(x)={\frac {1}{x}}$ , we should first remember

${\frac {d}{dx}}\ln(x)={\frac {1}{x}}$

Therefore, since ${\frac {1}{x}}$ is the derivative of $\ln(x)$ we can conclude that

$\int {\frac {dx}{x}}=\ln |x|+C$

Note that the polynomial integration rule does not apply when the exponent is $-1$ . This technique of integration must be used instead. Since the argument of the natural logarithm function must be positive (on the real line), the absolute value signs are added around its argument to ensure that the argument is positive.

Integral of the Exponential function

Since

${\frac {d}{dx}}e^{x}=e^{x}$

we see that $e^{x}$ is its own antiderivative. This allows us to find the integral of an exponential function:

$\int e^{x}dx=e^{x}+C$

Integral of Sine and Cosine

Recall that

${\frac {d}{dx}}\sin(x)=\cos(x)$

${\frac {d}{dx}}\cos(x)=-\sin(x)$

So $\sin(x)$ is an antiderivative of $\cos(x)$ and $-\cos(x)$ is an antiderivative of $\sin(x)$ . Hence we get the following rules for integrating $\sin(x)$ and $\cos(x)$

$\int \cos(x)dx=\sin(x)+C$
$\int \sin(x)dx=-\cos(x)+C$

Example

Suppose we want to integrate the function $f(x)=x^{4}+1+2\sin(x)$ . An application of the sum rule from above allows us to use the power rule and our rule for integrating $\sin(x)$ as follows,

$\int f(x)dx$	$=\int {\Big (}x^{4}+1+2\sin(x){\Big )}dx$
	$=\int x^{4}dx+\int 1\,dx+\int 2\sin(x)dx$
	$={\frac {x^{5}}{5}}+x-2\cos(x)+C$ .

Recognizing Derivatives and Reversing Derivative Rules

If we recognize a function $g(x)$ as being the derivative of a function $f(x)$ , then we can easily express the antiderivative of $g(x)$ :

$\int g(x)dx=f(x)+C$

For example, since

${\frac {d}{dx}}\sin(x)=\cos(x)$

we can conclude that

$\int \cos(x)dx=\sin(x)+C$

Similarly, since we know $e^{x}$ is its own derivative,

$\int e^{x}dx=e^{x}+C$

The power rule for derivatives can be reversed to give us a way to handle integrals of powers of $x$ . Since

${\frac {d}{dx}}x^{n}=nx^{n-1}$

we can conclude that

$\int nx^{n-1}dx=x^{n}+C$

or, a little more usefully,

$\int x^{n}dx={\frac {x^{n+1}}{n+1}}+C$

Resources

• Indefinite Integral, Wikibooks: Calculus
• Recognizing Derivatives and Substitution Rule, Wikibooks: Calculus
• Net Change Theorem, Definite Integral & Rates of Change Word Problems, Calculus by the Organic Chemistry Tutor