Properties of Definite Integrals

\(\int_a^a f(x) \,dx = 0\) (Equivalent Limits)

\(\int_a^b f(x) \,dx = -\int_b^a f(x) \,dx\) (Reversal of Limits)

\(\int_a^b k f(x) \,dx = k \int_a^b f(x) \,dx\) (Multiplication by a Constant)

\(\int_a^b f(x) \,dx = \int_a^c f(x) \,dx + \int_c^b f(x) \,dx\) (Adjacent Intervals, \(a < c < b\))

\(\int_a^b [f(x) + g(x)] \,dx = \int_a^b f(x) \,dx + \int_a^b g(x) \,dx\) (Addition)

\(\int_a^b [f(x) - g(x)] \,dx = \int_a^b f(x) \,dx - \int_a^b g(x) \,dx\) (Subtraction)

Basic Integration Formulas

\(\int e^x \,dx = e^x + C\)

\(\int a^x \,dx = \frac{1}{\ln a} a^x + C\)

\(\int \frac{1}{x} \,dx = \ln|x| + C\)

Power Rule for Integration

\(\int x^n \,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)

Trig Integrals

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

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

\(\int \sec^2 x \,dx = \tan x + C\)

\(\int \csc x \cot x \,dx = -\csc x + C\)

\(\int \sec x \tan x \,dx = \sec x + C\)

\(\int \csc^2 x \,dx = -\cot x + C\)

Inverse Trigonometric Integrals

\(\int \frac{1}{\sqrt{1-x^2}} \,dx = \sin^{-1} x + C\)

\(\int \frac{-1}{\sqrt{1-x^2}} \,dx = \cos^{-1} x + C\)

\(\int \frac{1}{1+x^2} \,dx = \tan^{-1} x + C\)

\(\int \frac{-1}{1+x^2} \,dx = \cot^{-1} x + C\)

\(\int \frac{1}{|x|\sqrt{x^2-1}} \,dx = \sec^{-1} x + C\)

\(\int \frac{-1}{|x|\sqrt{x^2-1}} \,dx = \csc^{-1} x + C\)