Mathematics

Table of Integrals

Integration rules and basic indefinite integrals: power rule, exponentials, logarithmic, and trigonometric functions.

  • Integral of a constant:
    ∫K dx=Kx+C\displaystyle \int K \, dx = Kx + C
  • Constant multiple rule:
    ∫K⋅f(x) dx=K⋅∫f(x) dx\displaystyle \int K \cdot f(x) \, dx = K \cdot \int f(x) \, dx
  • Sum rule:
    ∫[f(x)+g(x)] dx=∫f(x) dx+∫g(x) dx\displaystyle \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx
  • Integration by parts:
    ∫u⋅dv=u⋅v−∫v⋅du\displaystyle \int u \cdot dv = u \cdot v - \int v \cdot du
∫xm dx=xm+1m+1+C,(m≠−1)∫1x dx=ln⁡∣x∣+C∫1x+a dx=ln⁡∣x+a∣+C∫1x2 dx=−1x+C∫1x dx=2x+C∫ex dx=ex+C∫ax dx=axln⁡a+C∫sin⁡x dx=−cos⁡x+C∫cos⁡x dx=sin⁡x+C∫1cos⁡2x dx=tan⁡x+C∫1sin⁡2x dx=−cot⁡x+C∫11−x2 dx=arcsin⁡x+C∫1x2+1 dx=arctan⁡x+C∫1x2+a2 dx=1aarctan⁡xa+C\begin{aligned} & \int x^m \, dx = \frac{x^{m+1}}{m+1} + C, \quad (m \neq -1) \\[2ex] & \int \frac{1}{x} \, dx = \ln |x| + C \\[2ex] & \int \frac{1}{x+a} \, dx = \ln |x+a| + C \\[2ex] & \int \frac{1}{x^2} \, dx = -\frac{1}{x} + C \\[2ex] & \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C \\[2ex] & \int e^x \, dx = e^x + C \\[2ex] & \int a^x \, dx = \frac{a^x}{\ln a} + C \\[2ex] & \int \sin x \, dx = -\cos x + C \\[2ex] & \int \cos x \, dx = \sin x + C \\[2ex] & \int \frac{1}{\cos^2 x} \, dx = \tan x + C \\[2ex] & \int \frac{1}{\sin^2 x} \, dx = -\cot x + C \\[2ex] & \int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + C \\[2ex] & \int \frac{1}{x^2 + 1} \, dx = \arctan x + C \\[2ex] & \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan \frac{x}{a} + C \end{aligned}
*Solved by substituting xa=t\displaystyle \frac{x}{a} = t