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.