Common Derivatives and Integrals Reference Flashcards

Flashcards covering basic properties, common derivatives, basic integrals, inverse trigonometric functions, hyperbolic functions, miscellaneous integral forms, and standard integration techniques including u-substitution, integration by parts, trig substitutions, partial fractions, and products/quotients of trig functions.

Constant Multiple Rule for Indefinite Integrals

The integration rule stating that $\int c f(x)\,dx = c \int f(x)\,dx$, where $c$ is a constant.

Sum and Difference Rule for Indefinite Integrals

The integration rule stating that $\int (f(x) \pm g(x))\,dx = \int f(x)\,dx \pm \int g(x)\,dx$.

Fundamental Theorem of Calculus (Definite Integral Formula)

The rule stating that $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F(x) = \int f(x)\,dx$.

Constant Multiple Rule for Definite Integrals

The rule stating that $\int_a^b c f(x)\,dx = c \int_a^b f(x)\,dx$, where $c$ is a constant.

Sum and Difference Rule for Definite Integrals

The rule stating that $\int_a^b (f(x) \pm g(x))\,dx = \int_a^b f(x)\,dx \pm \int_a^b g(x)\,dx$.

Zero-Length Interval Integration Rule

The property stating that $\int_a^a f(x)\,dx = 0$.

Reversing Limits Property of Definite Integrals

The property stating that $\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$.

Additivity over Intervals Property

The property stating that $\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$.

Definite Integral of a Constant

The integration formula $\int_a^b c\,dx = c(b - a)$, where $c$ is a constant.

Non-negativity Property of Definite Integrals

The property stating that if $f(x) \ge 0$ on $a \le x \le b$, then $\int_a^b f(x)\,dx \ge 0$.

Dominance Property of Definite Integrals

The property stating that if $f(x) \ge g(x)$ on $a \le x \le b$, then $\int_a^b f(x)\,dx \ge \int_a^b g(x)\,dx$.

Integral of $dx$

The basic integral formula $\int dx = x + c$.

Integral of Constant $k$

The integral formula $\int k\,dx = kx + c$.

Power Rule for Integration

The rule $\int x^n\,dx = \frac{1}{n+1} x^{n+1} + c$, valid for $n \neq -1$.

Integration Power Rule Restriction

The explicit restriction $n \neq -1$ when evaluating $\int x^n\,dx = \frac{1}{n+1} x^{n+1} + c$.

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

The logarithmic integral formula $\int \frac{1}{x}\,dx = \ln(|x|) + c$.

Integral of $x^{-1}$

The logarithmic integral formula $\int x^{-1}\,dx = \ln(|x|) + c$.

Integral of Negative Power $x^{-n}$

The rule $\int x^{-n}\,dx = \frac{1}{-n+1} x^{-n+1} + c$, valid for $n \neq 1$.

Integration Negative Power Rule Restriction

The explicit restriction $n \neq 1$ when evaluating $\int x^{-n}\,dx = \frac{1}{-n+1} x^{-n+1} + c$.

Integral of Linear Denominator $\frac{1}{ax+b}$

The logarithmic formula $\int \frac{1}{ax+b}\,dx = \frac{1}{a}\ln(|ax+b|) + c$.

Integral of Rational Exponent $x^{p/q}$ (Unsimplified Form)

The formula $\int x^{p/q}\,dx = \frac{1}{\frac{p}{q}+1} x^{\frac{p}{q}+1} + c$.

Integral of Rational Exponent $x^{p/q}$ (Simplified Form)

The formula $\int x^{p/q}\,dx = \frac{q}{p+q} x^{\frac{p+q}{q}} + c$.

Integral of $\cos(u)$

The basic trigonometric integral $\int \cos(u)\,du = \sin(u) + c$.

Integral of $\sin(u)$

The basic trigonometric integral $\int \sin(u)\,du = -\cos(u) + c$.

Integral of $\sec^2(u)$

The trigonometric integral $\int \sec^2(u)\,du = \tan(u) + c$.

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

The trigonometric integral $\int \sec(u)\tan(u)\,du = \sec(u) + c$.

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

The trigonometric integral $\int \csc(u)\cot(u)\,du = -\csc(u) + c$.

Integral of $\csc^2(u)$

The trigonometric integral $\int \csc^2(u)\,du = -\cot(u) + c$.

Integral of $\tan(u)$ (Negative Cosine Form)

The integral formula $\int \tan(u)\,du = -\ln(|\cos(u)|) + c$.

Integral of $\tan(u)$ (Positive Secant Form)

The integral formula $\int \tan(u)\,du = \ln(|\sec(u)|) + c$.

Integral of $\cot(u)$ (Positive Sine Form)

The integral formula $\int \cot(u)\,du = \ln(|\sin(u)|) + c$.

Integral of $\cot(u)$ (Negative Cosecant Form)

The integral formula $\int \cot(u)\,du = -\ln(|\csc(u)|) + c$.

Integral of $\sec(u)$

The trigonometric integral $\int \sec(u)\,du = \ln(|\sec(u) + \tan(u)|) + c$.

Integral of $\sec^3(u)$

The trigonometric integral $\int \sec^3(u)\,du = \frac{1}{2}(\sec(u)\tan(u) + \ln(|\sec(u) + \tan(u)|)) + c$.

Integral of $\csc(u)$

The trigonometric integral $\int \csc(u)\,du = \ln(|\csc(u) - \cot(u)|) + c$.

Integral of $\csc^3(u)$

The trigonometric integral $\int \csc^3(u)\,du = \frac{1}{2}(-\csc(u)\cot(u) + \ln(|\csc(u) - \cot(u)|)) + c$.

Integral of $e^u$

The exponential integral formula $\int e^u\,du = e^u + c$.

Integral of $a^u$

The exponential integral formula $\int a^u\,du = \frac{a^u}{\ln(a)} + c$.

Integral of $\ln(u)$

The logarithmic integral formula $\int \ln(u)\,du = u \ln(u) - u + c$.

Integral of $e^{au}\sin(bu)$

The product formula $\int e^{au}\sin(bu)\,du = \frac{e^{au}}{a^2 + b^2}(a \sin(bu) - b \cos(bu)) + c$.

Integral of $u e^u$

The product formula $\int u e^u\,du = (u - 1)e^u + c$.

Integral of $e^{au}\cos(bu)$

The product formula $\int e^{au}\cos(bu)\,du = \frac{e^{au}}{a^2 + b^2}(a \cos(bu) + b \sin(bu)) + c$.

Integral of $\frac{1}{u \ln(u)}$

The logarithmic product integral $\int \frac{1}{u \ln(u)}\,du = \ln(|\ln(u)|) + c$.

Integral of $\frac{1}{\sqrt{a^2 - u^2}}$

The inverse trigonometric integral $\int \frac{1}{\sqrt{a^2 - u^2}}\,du = \sin^{-1}\left(\frac{u}{a}\right) + c$.

Integral of $\sin^{-1}(u)$

The inverse sine integral $\int \sin^{-1}(u)\,du = u \sin^{-1}(u) + \sqrt{1 - u^2} + c$.

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

The inverse trigonometric integral $\int \frac{1}{a^2 + u^2}\,du = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + c$.

Integral of $\tan^{-1}(u)$

The inverse tangent integral $\int \tan^{-1}(u)\,du = u \tan^{-1}(u) - \frac{1}{2}\ln(1 + u^2) + c$.

Integral of $\frac{1}{u\sqrt{u^2 - a^2}}$

The inverse trigonometric integral $\int \frac{1}{u\sqrt{u^2 - a^2}}\,du = \frac{1}{a} \sec^{-1}\left(\frac{u}{a}\right) + c$.

Integral of $\cos^{-1}(u)$

The inverse cosine integral $\int \cos^{-1}(u)\,du = u \cos^{-1}(u) - \sqrt{1 - u^2} + c$.

Integral of $\sinh(u)$

The hyperbolic integral formula $\int \sinh(u)\,du = \cosh(u) + c$.

Integral of \sech(u)\tanh(u)

The hyperbolic integral formula \int \sech(u)\tanh(u)\,du = -\sech(u) + c.

Integral of \sech^2(u)

The hyperbolic integral formula \int \sech^2(u)\,du = \tanh(u) + c.

Integral of $\cosh(u)$

The hyperbolic integral formula $\int \cosh(u)\,du = \sinh(u) + c$.

Integral of \csch(u)\coth(u)

The hyperbolic integral formula \int \csch(u)\coth(u)\,du = -\csch(u) + c.

Integral of \csch^2(u)

The hyperbolic integral formula \int \csch^2(u)\,du = -\coth(u) + c.

Integral of $\tanh(u)$

The hyperbolic integral formula $\int \tanh(u)\,du = \ln(\cosh(u)) + c$.

Integral of \sech(u)

The hyperbolic integral formula \int \sech(u)\,du = \tan^{-1}(|\sinh(u)|) + c.

Integral of $\frac{1}{a^2 - u^2}$

The rational integral formula $\int \frac{1}{a^2 - u^2}\,du = \frac{1}{2a} \ln\left(\left|\frac{u+a}{u-a}\right|\right) + c$.

Integral of $\sqrt{a^2 + u^2}$

The radical integral formula $\int \sqrt{a^2 + u^2}\,du = \frac{u}{2}\sqrt{a^2 + u^2} + \frac{a^2}{2} \ln(|u + \sqrt{a^2 + u^2}|) + c$.

Integral of $\frac{1}{u^2 - a^2}$

The rational integral formula $\int \frac{1}{u^2 - a^2}\,du = \frac{1}{2a} \ln\left(\left|\frac{u-a}{u+a}\right|\right) + c$.

Integral of $\sqrt{u^2 - a^2}$

The radical integral formula $\int \sqrt{u^2 - a^2}\,du = \frac{u}{2}\sqrt{u^2 - a^2} - \frac{a^2}{2} \ln(|u + \sqrt{u^2 - a^2}|) + c$.

Integral of $\sqrt{a^2 - u^2}$

The radical integral formula $\int \sqrt{a^2 - u^2}\,du = \frac{u}{2}\sqrt{a^2 - u^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{u}{a}\right) + c$.

Integral of $\sqrt{2au - u^2}$

The radical integral formula $\int \sqrt{2au - u^2}\,du = \frac{u-a}{2}\sqrt{2au - u^2} + \frac{a^2}{2} \cos^{-1}\left(\frac{a-u}{a}\right) + c$.

Definite Integral $u$-Substitution Rule

The rule stating $\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$ using substitution $u = g(x)$.

$u$-Substitution Differential Relation

The differential equation $du = g'(x)\,dx$ corresponding to substitution $u = g(x)$.

Indefinite Integral $u$-Substitution Rule

The instruction to drop limits of integration when applying $u$-substitution to an indefinite integral.

Integration by Parts Formula (Indefinite)

The integration technique formula $\int u\,dv = uv - \int v\,du$.

Integration by Parts Formula (Definite)

The definite integration formula $\int_a^b u\,dv = uv|_a^b - \int_a^b v\,du$.

Computing $du$ in Integration by Parts

The step in integration by parts where $du$ is computed by differentiating the chosen $u$.

Computing $v$ in Integration by Parts

The step in integration by parts where $v$ is computed using $v = \int dv$.

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

The substitution $x = \frac{a}{b}\sin(\theta)$.

Trigonometric Identity for $\sqrt{a^2 - b^2 x^2}$

The identity $\cos^2(\theta) = 1 - \sin^2(\theta)$ paired with substitution $x = \frac{a}{b}\sin(\theta)$.

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

The substitution $x = \frac{a}{b}\sec(\theta)$.

Trigonometric Identity for $\sqrt{b^2 x^2 - a^2}$

The identity $\tan^2(\theta) = \sec^2(\theta) - 1$ paired with substitution $x = \frac{a}{b}\sec(\theta)$.

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

The substitution $x = \frac{a}{b}\tan(\theta)$.

Trigonometric Identity for $\sqrt{a^2 + b^2 x^2}$

The identity $\sec^2(\theta) = 1 + \tan^2(\theta)$ paired with substitution $x = \frac{a}{b}\tan(\theta)$.

Degree Requirement for Partial Fractions

The requirement that for $\int \frac{P(x)}{Q(x)}\,dx$, the degree (largest exponent) of $P(x)$ must be smaller than the degree of $Q(x)$.

Initial Step in Partial Fraction Integration

Factoring the denominator $Q(x)$ as completely as possible before setting up the partial fraction decomposition.

Partial Fractions Decomposition Setup Table

The rules governing the decomposition terms in P.F.D. based on factors of $Q(x)$.

Partial Fraction Decomposition Term for $ax + b$

The single term $\frac{A}{ax + b}$ added for factor $ax + b$ of $Q(x)$.

Partial Fraction Decomposition Term for $(ax + b)^k$

The sum of terms $\frac{A_1}{ax + b} + \frac{A_2}{(ax + b)^2} + \cdots + \frac{A_k}{(ax + b)^k}$ added for factor $(ax + b)^k$ of $Q(x)$.

Partial Fraction Decomposition Term for $ax^2 + bx + c$

The single term $\frac{Ax + B}{ax^2 + bx + c}$ added for irreducible quadratic factor $ax^2 + bx + c$ of $Q(x)$.

Partial Fraction Decomposition Term for $(ax^2 + bx + c)^k$

The sum of terms $\frac{A_1 x + B_1}{ax^2 + bx + c} + \cdots + \frac{A_k x + B_k}{(ax^2 + bx + c)^k}$ added for repeated irreducible quadratic factor $(ax^2 + bx + c)^k$ of $Q(x)$.

Strategy for $\int \sin^n(x)\cos^m(x)\,dx$ with $n$ Odd

Strip 1 sine out and convert the rest to cosines using $\sin^2(x) = 1 - \cos^2(x)$.

Substitution for $\int \sin^n(x)\cos^m(x)\,dx$ with $n$ Odd

Use the substitution $u = \cos(x)$.

Strategy for $\int \sin^n(x)\cos^m(x)\,dx$ with $m$ Odd

Strip 1 cosine out and convert the rest to sines using $\cos^2(x) = 1 - \sin^2(x)$.

Substitution for $\int \sin^n(x)\cos^m(x)\,dx$ with $m$ Odd

Use the substitution $u = \sin(x)$.

Strategy for $\int \sin^n(x)\cos^m(x)\,dx$ when Both $n$ and $m$ are Odd

Use either the strategy for $n$ odd (strip 1 sine) or $m$ odd (strip 1 cosine).

Strategy for $\int \sin^n(x)\cos^m(x)\,dx$ when Both $n$ and $m$ are Even

Use double angle and/or half angle formulas to reduce the integral into an integrable form.

Strategy for $\int \tan^n(x)\sec^m(x)\,dx$ with $n$ Odd

Strip 1 tangent and 1 secant out and convert the rest to secants using $\tan^2(x) = \sec^2(x) - 1$.

Substitution for $\int \tan^n(x)\sec^m(x)\,dx$ with $n$ Odd

Use the substitution $u = \sec(x)$.

Strategy for $\int \tan^n(x)\sec^m(x)\,dx$ with $m$ Even

Strip 2 secants out and convert the rest to tangents using $\sec^2(x) = 1 + \tan^2(x)$.

Substitution for $\int \tan^n(x)\sec^m(x)\,dx$ with $m$ Even

Use the substitution $u = \tan(x)$.

Strategy for $\int \tan^n(x)\sec^m(x)\,dx$ with $n$ Odd and $m$ Even

Use either the strategy for $n$ odd or $m$ even.

Strategy for $\int \tan^n(x)\sec^m(x)\,dx$ with $n$ Even and $m$ Odd

Each integral in this case will be dealt with differently.

Conversion Example of $\cos^6(x)$

The algebraic conversion $\cos^6(x) = (\cos^2(x))^3 = (1 - \sin^2(x))^3$.

Constant of Integration

The arbitrary constant $c$ appended to antiderivatives in indefinite integrals.

Lower Limit of Integration

The value $a$ evaluated at the lower boundary in $\int_a^b f(x)\,dx$.

Upper Limit of Integration

The value $b$ evaluated at the upper boundary in $\int_a^b f(x)\,dx$.

Rational Expression in Integration

An expression $\frac{P(x)}{Q(x)}$ formed by dividing polynomial $P(x)$ by polynomial $Q(x)$.