$\int f(x) dx = F(x) + C$, where $F'(x) = f(x)$

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

$\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

$\int c \cdot f(x) dx = c \int f(x) dx$

$-\cos(x) + C$

$\sin(x) + C$

$\tan(x) + C$

$-\cot(x) + C$

$\sec(x) + C$

$-\csc(x) + C$

$\ln|x| + C$

$e^x + C$

Because the derivative of a constant is zero, any constant could be part of the original function. '+C' represents all possible constants.

To find the antiderivative of $x^n$, add 1 to the exponent and divide by the new exponent, then add the constant of integration, C.

Because if $n = -1$, the denominator becomes zero, making the expression undefined.

The antiderivative of a sum of functions is the sum of their individual antiderivatives. This simplifies integration of complex expressions.

A constant factor can be moved outside the integral sign, simplifying the integration process.

The natural logarithm is only defined for positive values. The absolute value ensures that the antiderivative is defined for all non-zero x.

Antiderivatives are the inverse operation of derivatives. Finding the antiderivative reverses the process of differentiation.

Apply the reverse power rule to each term of the polynomial, then add the constant of integration, C.

Differentiate the antiderivative. If the result is the original function, the antiderivative is correct.

Definite integrals have upper and lower bounds and result in a numerical value. Indefinite integrals do not have bounds and result in a function plus a constant.

An integral without specified bounds, resulting in a family of functions that differ by a constant.

A constant term, denoted as 'C', added to the antiderivative to account for all possible antiderivatives.

A function whose derivative is equal to a given function.

A method for finding the antiderivative of power functions.

A set of functions that share the same derivative but differ by a constant.

The integral of a sum of functions is the sum of their individual integrals.

The integral of a constant times a function is the constant times the integral of the function.

$\ln|x| + C$

$e^x + C$

$-\cos(x) + C$