Calculus: Integration Formulas and Techniques

A complete set of vocabulary flashcards covering basic trigonometric, logarithmic, exponential, inverse tangent, and special algebraic integration formulas and example problems from the lecture notes.

Integral of $e^x$

The standard exponential rule where the integral of the function $e^x$ is $e^x$.

Integral of $e^{f(x)}$

The exponential function rule presented as having the integral equal to $\frac{e^{f(x)}}{f'(x)}$.

Integral of $\frac{1}{1+x^2}$

The basic inverse trigonometric integral resulting in $\tan^{-1}(x)$.

Integral of $\frac{1}{a^2+x^2}$

The inverse tangent formula for a squared constant $a^2$ in the denominator, resulting in $\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right)$.

Integral of $\frac{1}{a^2+(bx)^2}$

The generalized inverse tangent formula accounting for the linear coefficient $b$, given as $\frac{1}{a}\tan^{-1}\left(\frac{bx}{a}\right) \times \frac{1}{b}$.

Integral of $\sin(x)$

The basic trigonometric integral of $\sin(x)$, which equals $-\cos(x)$.

Integral of $\cos(x)$

The basic trigonometric integral of $\cos(x)$, which equals $\sin(x)$.

Integral of $\sec^2(x)$

The trigonometric integral of $\sec^2(x)$, which equals $\tan(x)$.

Integral of $\csc(x)\cot(x)$

The trigonometric integral of $\csc(x)\cot(x)$, which equals $-\csc(x)$.

Integral of $\sec(x)\tan(x)$

The trigonometric integral of $\sec(x)\tan(x)$, which equals $\sec(x)$.

Integral of $\csc^2(x)$

The trigonometric integral of $\csc^2(x)$, which equals $-\cot(x)$.

Integral of $\frac{1}{x}$

The basic logarithmic function rule where the integral is $\ln(x)$.

Logarithmic Rule $\frac{f'(x)}{f(x)}$

An important integration rule stating that when the derivative of the denominator appears in the numerator, the integral is $\ln(f(x))$.

Special Log Formula for $\frac{1}{a^2-x^2}$

An algebraic integration formula for a difference of squares in the denominator yielding $\frac{1}{2a}\ln\left(\frac{a+x}{a-x}\right)$.

Special Log Formula for $\frac{1}{x^2-a^2}$

An algebraic integration formula for a difference of squares in the denominator yielding $\frac{1}{2a}\ln\left(\frac{x-a}{x+a}\right)$.

Evaluation of $\int \frac{3}{1+x^2}\,dx$

An application of the basic inverse tangent rule pulling out constant $3$ to evaluate to $3\tan^{-1}(x)$.

Evaluation of $\int \frac{5}{3+x^2}\,dx$

An application of the inverse tangent formula rewriting the denominator as $(\sqrt{3})^2+x^2$ to yield $\frac{5}{\sqrt{3}}\tan^{-1}\left(\frac{x}{\sqrt{3}}\right)$.

Evaluation of $\int \frac{4}{9-x^2}\,dx$

An application of the special log formula rewritten as $(3)^2-(x)^2$ yielding $4 \times \frac{1}{2(3)}\ln\left(\frac{3+x}{3-x}\right) = \frac{2}{3}\ln\left(\frac{3+x}{3-x}\right)$.

Evaluation of $\int \frac{3}{x^2-16}\,dx$

An application of the special log formula rewritten as $(x)^2-(4)^2$ yielding $3 \times \frac{1}{2(4)}\ln\left(\frac{x-4}{x+4}\right) = \frac{3}{8}\ln\left(\frac{x-4}{x+4}\right)$.

Evaluation of $\int \frac{x}{4+x^2}\,dx$

An application of the $\frac{f'(x)}{f(x)}$ rule where multiplying and dividing by $2$ yields $\frac{1}{2}\ln(4+x^2)$.

Evaluation of $\int \frac{3}{4+x^2}\,dx$

An application of the inverse tangent formula with a constant numerator yielding $3 \times \frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) = \frac{3}{2}\tan^{-1}\left(\frac{x}{2}\right)$.