Integral

INTEGRAL

Integration is the calculation of an integral. It involves finding the antiderivative of a function. Integrals in mathematics are used to find various quantities, such as areas, volumes, and displacements.

We can categorize the Integrals into two main types:

Definite Integrals:
Indefinite Integrals:

DEFINITE INTEGRAL

These have limits or boundaries and yield a numerical answer. For example, finding the area under a curve between two points is a definite integral because it involves boundaries.

\int_{a}^{b} f(x) \, dx = f(b) – f(a)

INDEFINITE INTEGRAL

These do not have limits or boundaries and yield an algebraic answer. These involve finding antiderivatives, which means undoing the process of differentiation.

\int f(x) \, dx + C

INTEGRAL FORMULA

BASIC INTEGRAL

\int k \, dx = kx + C

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

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

TRIGONOMETRIC INTEGRAL

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

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

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

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

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

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

$\int e^x \, dx = e^x + C$

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

EXPONENTIAL AND LOGARITHMIC INTEGRAL

$\int e^x \, dx = e^x + C$

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

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

INVERSE TRIGONOMETRIC INTEGRAL

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

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

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

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

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

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