Integration One Page Complete Revision

Complete revision flashcards for Integration, covering basic formulas, common integrals, methods like Substitution and Integration by Parts, and standard substitutions.

$$\int k \, dx$$

$$kx + C$$

$$\int x^n \, dx$$

$$\frac{x^{n+1}}{n+1} + C, n \neq -1$$

$$\int \frac{1}{x} \, dx$$

$$\ln|x| + C$$

$$\int e^x \, dx$$

$$e^x + C$$

$$\int a^x \, dx$$

$$\frac{a^x}{\ln(a)} + C, a > 0, a \neq 1$$

$$\int \sin(x) \, dx$$

$$-\cos(x) + C$$

$$\int \cos(x) \, dx$$

$$\sin(x) + C$$

$$\int \sec^2(x) \, dx$$

$$\tan(x) + C$$

$$\int \csc^2(x) \, dx$$

$$-\cot(x) + C$$

$$\int \sec(x)\tan(x) \, dx$$

$$\sec(x) + C$$

$$\int \csc(x)\cot(x) \, dx$$

$$-\csc(x) + C$$

$$\int \frac{1}{1+x^2} \, dx$$

$$\tan^{-1}(x) + C$$

$$\int \frac{1}{\sqrt{a^2-x^2}} \, dx$$

$$\sin^{-1}\left(\frac{x}{a}\right) + C$$

Substitution Method

A method where we let $I = \int f(x) \, dx$, put $x = g(t) \Rightarrow dx = g'(t) \, dt$, resulting in $I = \int f(g(t)) g'(t) \, dt$.

Integration by Parts

$\int u \, dv = uv - \int v \, du$, where $u$ is chosen as differentiable and $dv$ as integrable.

Trigonometric Integration

Used for integrals containing expressions like $\sqrt{a^2-x^2}$, $\sqrt{a^2+x^2}$, or $\sqrt{x^2-a^2}$ with suitable substitutions.

Partial Fractions

Used when the integrand is of the form $\frac{P(x)}{Q(x)}$ where $Q(x)$ can be factored.

$$\frac{d}{dx}\left(\int f(x) \, dx\right)$$

$$f(x)$$

$$\int \frac{f'(x)}{f(x)} \, dx$$

$$\ln|f(x)| + C$$

$$\int e^{ax} \, dx$$

$$\frac{e^{ax}}{a} + C$$

$$\int \sin(ax) \, dx$$

$$-\frac{\cos(ax)}{a} + C$$

$$\int \frac{1}{x^2 + a^2} \, dx$$

$$\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$$

$$\int \frac{1}{\sqrt{x^2 \pm a^2}} \, dx$$

$$\ln|x + \sqrt{x^2 \pm a^2}| + C$$

$$\int \frac{1}{x^2 - a^2} \, dx$$

$$\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C$$

Standard Substitution for $\sqrt{a^2 - x^2}$

$$x = a \sin(\theta)$$

Standard Substitution for $\sqrt{a^2 + x^2}$

$$x = a \tan(\theta)$$

Standard Substitution for $\sqrt{x^2 - a^2}$

$$x = a \sec(\theta)$$

Constant of Integration (+C)

Values that must be added to all indefinite integrals.