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.

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: $\frac{2}{3}x^{3/2} - x^{-1} + C$.

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

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

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

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

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

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

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

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

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

$-\cos(x) + C$

$\sin(x) + C$

$\tan(x) + C$

$-\cot(x) + C$

$\sec(x) + C$

$-\csc(x) + C$

$\ln|x| + C$

$e^x + C$