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.

$\ln|x| + C$

$e^x + C$

$-\cos(x) + C$

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

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

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

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

$-\cos(x) + C$

$\sin(x) + C$

$\tan(x) + C$

$-\cot(x) + C$

$\sec(x) + C$

$-\csc(x) + C$

$\ln|x| + C$

$e^x + C$

Apply the sums rule: $\int 3x^2 dx + \int 2x dx + \int 1 dx$. Apply the reverse power rule to each term: $x^3 + x^2 + x + C$.

Apply the multiples rule: $5\int \cos(x) dx$. Integrate: $5\sin(x) + C$.

Apply the sums rule: $\int e^x dx - \int \sin(x) dx$. Integrate each term: $e^x + \cos(x) + C$.

Apply the multiples rule: $4\int \frac{1}{x} dx$. Integrate: $4\ln|x| + C$.

Rewrite: $\int (x^{1/2} + x^{-2}) dx$. Apply the reverse power rule to each term: $\frac{2}{3}x^{3/2} - x^{-1} + C$.

$\int x^3 dx + \int 5 dx = \frac{x^4}{4} + 5x + C$

$2\int \sin(x) dx - 3\int \cos(x) dx = -2\cos(x) - 3\sin(x) + C$

$\int x^{-3} dx + \int e^x dx = -\frac{1}{2x^2} + e^x + C$

$\int 7x^6 dx - \int \frac{2}{x} dx = x^7 - 2\ln|x| + C$

$\int 5x^{-1/2} dx - \int 4x^3 dx = 10\sqrt{x} - x^4 + C$