# Integration Rules Reference Page

• Sum/Difference Rule:
$$\int [f(x) \pm g(x)] \; dx= \int f(x) \; dx \pm \int g(x) \; dx$$
• Constant Multiplier Rule:
$$\int k \cdot f(x) \; dx= k \int f(x)\; dx$$
Let $$C$$ be a constant.

• $$\displaystyle \int 0 \; dx = C$$
• $$\displaystyle \int \; dx = x + C$$
• $$\displaystyle \int x^n \; dx= \begin{cases} \dfrac{x^{n+1}}{n+1} +C , & \; n \not = -1 \\ \\ \ln |x| + C, & \; n = -1 \end{cases}$$
• $$\displaystyle \int \cos x \; dx = \sin x + C$$
• $$\displaystyle \int \sin x \; dx = -\cos x + C$$
• $$\displaystyle \int \tan x \; dx = – \ln |\cos x| + C$$
• $$\displaystyle \int \cot x \; dx = \ln |\sin x| + C$$
• $$\displaystyle \int \sec^2 x \; dx = \tan x + C$$
• $$\displaystyle \int \csc^2 x \; dx = -\cot x + C$$
• $$\displaystyle \int \sec x \tan x \; dx = \sec x + C$$
• $$\displaystyle \int \csc x \cot x \; dx = -\csc x + C$$
• $$\displaystyle \int \csc x \; dx = -\ln|\csc x+\cot x| + C$$
• $$\displaystyle \int \sec x \; dx = \ln|\sec x + \tan x| + C$$
• $$\displaystyle \int e^x \; dx = e^x+ C$$
• $$\displaystyle \int a^x \; dx = \dfrac{a^x}{\ln a} + C, \; \; a>0, a \not = 1$$
Let $$g$$ and $$u$$ be functions such that $$f(x) = g(u(x))$$ (That is, $$f(x)$$ can be written as a composition of $$g$$ with $$u$$) and assume that the antiderivative $$G(x)$$ of $$g$$ is known then
\begin{aligned} \int f(x) \; dx & = \int g(u(x)) \dfrac{du}{dx} \; dx \\ & = \int g(u) \; du = G(u) + C. \end{aligned}
The Fundamental Theorem of Calculus (Part 1): If $$f(x)$$ is continuous on $$[a,b]$$ and $$F(x)$$ is an antiderivative of $$f(x)$$ then
$$\int_a^b f(x) \; dx = F(x)\Big|_a^b$$
The Fundamental Theorem of Calculus (Part 2): If $$f$$ is continuous on $$[a,b]$$ then
$$\frac{d}{dx} \left[\int_a^x f(t) \; dt \right] = f(x).$$
The average value of $$f(x)$$ on $$[a,b]$$ is denoted $$f_{avg}$$ and is given by
$$f_{avg} = \frac{1}{b-a}\int_a^b f(x)\; dx.$$
The definite integral of $$f(x)$$ on $$[a,b]$$ is denoted
$$\int_a^b f(x) \; dx$$
and informally represents the signed “area” of the region “bounded” by the graphs of $$y = f(x)$$, there vertical lines $$x=a$$ and $$x =b$$, and the $$x$$-axis.
Right Rectangular Rule If $$f(x)$$ is continuous on $$[a,b]$$,
$$\int_a^b f(x) \; dx \approx \sum_{k = 1}^n f(a+k \Delta x) \Delta x$$
and
$$\int_a^b f(x) \; dx =\lim_{n \rightarrow \infty} \sum_{k = 1}^n f(a+k \Delta x) \Delta x$$
where $$\Delta x = \frac{b-a}{n}.$$
Left Rectangular Rule If $$f(x)$$ is continuous on $$[a,b]$$,
$$\int_a^b f(x) \; dx \approx \sum_{k = 1}^n f(a+(k-1) \Delta x) \Delta x$$
and
$$\int_a^b f(x) \; dx =\lim_{n \rightarrow \infty} \sum_{k = 1}^n f(a+(k-1) \Delta x) \Delta x$$
where $$\Delta x = \frac{b-a}{n}.$$
Midpoint Rule If $$f(x)$$ is continuous on $$[a,b]$$,
$$\int_a^b f(x) \; dx \approx \sum_{k = 1}^n f\left(a+\left(k-\frac{1}{2} \right) \Delta x \right) \Delta x$$
and
$$\int_a^b f(x) \; dx =\lim_{n \rightarrow \infty} \sum_{k = 1}^n f\left(a+\left(k-\frac{1}{2}\right) \Delta x \right) \Delta x$$
where $$\Delta x = \frac{b-a}{n}.$$