Key Points

Integration is the inverse process of differentiation. If the derivative of a function $F(x)$ is $f(x)$, then the indefinite integral of $f(x)$ is given by $\int f(x) dx = F(x) + C$, where $C$ is the constant of integration.

The power rule for integration is $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ for any real number $n \neq -1$. Other key formulas are $\int \frac{1}{x} dx = \log|x| + C$ and $\int e^x dx = e^x + C$.

Fundamental trigonometric integrals include $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$, $\int \sec^2 x dx = \tan x + C$, and $\int \csc^2 x dx = -\cot x + C$.

This method transforms an integral by changing the variable. To solve $\int f(g(x))g'(x) dx$, substitute $t = g(x)$, which implies $dt = g'(x) dx$. The integral simplifies to $\int f(t) dt$.

This method is used to integrate the product of two functions. The formula is $\int u v dx = u \int v dx - \int (\frac{du}{dx} \int v dx) dx$. The choice of the first function ($u$) and second function ($v$) is critical for simplification.

A useful shortcut formula is $\int e^x [f(x) + f'(x)] dx = e^x f(x) + C$. To apply this, identify a function $f(x)$ and its derivative $f'(x)$ within the integrand.

This method applies to proper rational functions $\frac{P(x)}{Q(x)}$. The function is decomposed into a sum of simpler fractions based on the factors of the denominator $Q(x)$. For example, $\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$.

Three standard forms are $\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log|\frac{x-a}{x+a}| + C$, $\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log|\frac{a+x}{a-x}| + C$, and $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}(\frac{x}{a}) + C$.

Key formulas include $\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C$ and $\int \frac{dx}{\sqrt{x^2 \pm a^2}} = \log|x + \sqrt{x^2 \pm a^2}| + C$.

Important formulas are $\int \sqrt{a^2 - x^2} dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}(\frac{x}{a}) + C$ and $\int \sqrt{x^2 \pm a^2} dx = \frac{x}{2}\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\log|x + \sqrt{x^2 \pm a^2}| + C$.

For integrals involving a general quadratic expression like $ax^2+bx+c$ in the denominator, convert it into a sum or difference of two squares to match standard integral forms.

To solve integrals like $\int \frac{px+q}{ax^2+bx+c} dx$, express the numerator as $px+q = A \frac{d}{dx}(ax^2+bx+c) + B$. This splits the integral into two manageable parts.

This theorem connects differentiation and integration. To evaluate a definite integral $\int_a^b f(x) dx$, find an anti-derivative $F(x)$ of $f(x)$, and the value is $F(b) - F(a)$.

Key properties are $\int_a^b f(x) dx = -\int_b^a f(x) dx$ and $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$. The variable is a dummy, so $\int_a^b f(x) dx = \int_a^b f(t) dt$.

A powerful property for simplification is $\int_0^a f(x) dx = \int_0^a f(a-x) dx$. The general form is $\int_a^b f(x) dx = \int_a^b f(a+b-x) dx$.

For a symmetric interval $[-a, a]$: if $f(x)$ is even ($f(-x) = f(x)$), then $\int_{-a}^a f(x) dx = 2\int_0^a f(x) dx$. If $f(x)$ is odd ($f(-x) = -f(x)$), then $\int_{-a}^a f(x) dx = 0$.

When using substitution in a definite integral, the limits of integration must be changed according to the substitution. If $t=g(x)$, the new limits correspond to the values of $g(x)$ at the original limits.

These are standard results: $\int \tan x dx = \log|\sec x| + C$, $\int \cot x dx = \log|\sin x| + C$, $\int \sec x dx = \log|\sec x + \tan x| + C$, and $\int \csc x dx = \log|\csc x - \cot x| + C$.