Common Derivatives and Integrals

Common Derivatives and Integrals
Derivatives

  • Basic Properties/Formulas/Rules

    • ddx[cf(x)]=cddx[f(x)]c is a constant\frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)] \quad c \text{ is a constant}

    • ddx[f(x)±g(x)]=ddx[f(x)]±ddx[g(x)]\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]

    • ddx[xn]=nxn−1,n is any number\frac{d}{dx}[x^n] = nx^{n-1}, \quad n \text{ is any number}

    • ddx[c]=0,c is a constant\frac{d}{dx}[c] = 0, \quad c \text{ is a constant}

    • Product Rule: ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

    • Quotient Rule: ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)(g(x))2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

    • Chain Rule: ddx[f(g(x))]=f′(g(x))g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)

  • Common Derivatives

    • Polynomials

    • ddx[c]=0\frac{d}{dx}[c] = 0

    • ddx[x]=1\frac{d}{dx}[x] = 1

    • ddx[cxn]=ncxn−1\frac{d}{dx}[cx^n] = ncx^{n-1}

    • Trigonometric Functions

    • ddx[sin⁡(x)]=cos⁡(x)\frac{d}{dx}[\sin(x)] = \cos(x)

    • ddx[cos⁡(x)]=−sin⁡(x)\frac{d}{dx}[\cos(x)] = -\sin(x)

    • ddx[tan⁡(x)]=sec⁡2(x)\frac{d}{dx}[\tan(x)] = \sec^2(x)

    • ddx[csc⁡(x)]=−csc⁡(x)cot⁡(x)\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)

    • ddx[sec⁡(x)]=sec⁡(x)tan⁡(x)\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)

    • ddx[cot⁡(x)]=−csc⁡2(x)\frac{d}{dx}[\cot(x)] = -\csc^2(x)

    • Inverse Trigonometric Functions

    • ddx[sin⁡−1(x)]=11−x2\frac{d}{dx}[\sin^{-1}(x)] = \frac{1}{\sqrt{1-x^2}}

    • ddx[cos⁡−1(x)]=−11−x2\frac{d}{dx}[\cos^{-1}(x)] = -\frac{1}{\sqrt{1-x^2}}

    • ddx[tan⁡−1(x)]=11+x2\frac{d}{dx}[\tan^{-1}(x)] = \frac{1}{1+x^2}

    • ddx[csc⁡−1(x)]=−1∣x∣x2−1\frac{d}{dx}[\csc^{-1}(x)] = -\frac{1}{|x|\sqrt{x^2-1}}

    • ddx[sec⁡−1(x)]=1∣x∣x2−1\frac{d}{dx}[\sec^{-1}(x)] = \frac{1}{|x|\sqrt{x^2-1}}

    • ddx[cot⁡−1(x)]=−11+x2\frac{d}{dx}[\cot^{-1}(x)] = -\frac{1}{1+x^2}

    • Exponential & Logarithmic Functions

    • ddx[ax]=axln⁡(a)\frac{d}{dx}[a^x] = a^x \ln(a)

    • ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

    • \frac{d}{dx}[\ln(x)] = \frac{1}{x}, \quad x > 0

    • ddx[ln⁡∣x∣]=1x,x≠0\frac{d}{dx}[\ln|x|] = \frac{1}{x}, \quad x \neq 0

    • \frac{d}{dx}[\log_a(x)] = \frac{1}{x \ln(a)}, \quad x > 0

    • Hyperbolic Functions

    • ddx[sinh⁡(x)]=cosh⁡(x)\frac{d}{dx}[\sinh(x)] = \cosh(x)

    • ddx[cosh⁡(x)]=sinh⁡(x)\frac{d}{dx}[\cosh(x)] = \sinh(x)

    • \frac{d}{dx}[\tanh(x)] = \sech^2(x)

    • \frac{d}{dx}[\csch(x)] = -\csch(x)\coth(x)

    • \frac{d}{dx}[\sech(x)] = -\sech(x)\tanh(x)

    • \frac{d}{dx}[\coth(x)] = -\csch^2(x)

Integrals

  • Basic Properties/Formulas/Rules

    • ∫cf(x)dx=c∫f(x)dx,c is a constant\int cf(x)dx = c \int f(x)dx, \quad c \text{ is a constant}

    • ∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\int [f(x) \pm g(x)]dx = \int f(x)dx \pm \int g(x)dx

    • ∫abf(x)dx=F(b)−F(a) where F(x)=∫f(x)dx\int_a^b f(x)dx = F(b) - F(a) \text{ where } F(x) = \int f(x)dx

    • ∫<em>abcf(x)dx=c∫</em>abf(x)dx,c is a constant\int<em>a^b cf(x)dx = c \int</em>a^b f(x)dx, \quad c \text{ is a constant}

    • ∫<em>abf(x)±g(x)dx=∫</em>abf(x)dx±∫abg(x)dx\int<em>a^b f(x) \pm g(x)dx = \int</em>a^b f(x)dx \pm \int_a^b g(x)dx

    • ∫aaf(x)dx=0\int_a^a f(x)dx = 0

    • ∫<em>abf(x)dx=−∫</em>baf(x)dx\int<em>a^b f(x)dx = -\int</em>b^a f(x)dx

    • ∫<em>abf(x)dx=∫</em>caf(x)dx+∫cbf(x)dx\int<em>a^b f(x)dx = \int</em>c^a f(x)dx + \int_c^b f(x)dx

    • ∫abcdx=c(b−a),c is a constant\int_a^b c dx = c(b-a), \quad c \text{ is a constant}

    • If f(x)≥0f(x) \geq 0 on a≤x≤ba \leq x \leq b then ∫abf(x)dx≥0\int_a^b f(x)dx \geq 0

    • If f(x)≥g(x)f(x) \geq g(x) on a≤x≤ba \leq x \leq b then ∫<em>abf(x)dx≥∫</em>abg(x)dx\int<em>a^b f(x)dx \geq \int</em>a^b g(x)dx

  • Common Integrals

    • Polynomials

    • ∫dx=x+c\int dx = x + c

    • ∫kdx=kx+c\int k dx = kx + c

    • ∫xndx=1n+1xn+1+c,n≠−1\int x^n dx = \frac{1}{n+1}x^{n+1} + c, \quad n \neq -1

    • ∫1xdx=ln⁡∣x∣+c\int \frac{1}{x}dx = \ln|x| + c

    • ∫x−1dx=ln⁡∣x∣+c\int x^{-1}dx = \ln|x| + c

    • ∫x−ndx=1−n+1x−n+1+c,n≠1\int x^{-n}dx = \frac{1}{-n+1}x^{-n+1} + c, \quad n \neq 1

    • ∫1ax+bdx=1aln⁡∣ax+b∣+c\int \frac{1}{ax+b}dx = \frac{1}{a} \ln|ax+b| + c

    • ∫xpqdx=1pq+1xp+q+c\int x^p q dx = \frac{1}{pq+1} x^{p+q} + c

    • Trigonometric Functions

    • ∫cos⁡(u)du=sin⁡(u)+c\int \cos(u) du = \sin(u) + c

    • ∫sin⁡(u)du=−cos⁡(u)+c\int \sin(u) du = -\cos(u) + c

    • ∫sec⁡2(u)du=tan⁡(u)+c\int \sec^2(u) du = \tan(u) + c

    • ∫sec⁡(u)tan⁡(u)du=sec⁡(u)+c\int \sec(u)\tan(u) du = \sec(u) + c

    • ∫csc⁡(u)cot⁡(u)du=−csc⁡(u)+c\int \csc(u)\cot(u) du = -\csc(u) + c

    • ∫csc⁡2(u)du=−cot⁡(u)+c\int \csc^2(u) du = -\cot(u) + c

    • ∫tan⁡(u)du=−ln⁡∣cos⁡(u)∣+c=ln⁡∣sec⁡(u)∣+c\int \tan(u) du = -\ln|\cos(u)| + c = \ln|\sec(u)| + c

    • ∫cot⁡(u)du=ln⁡∣sin⁡(u)∣+c=−ln⁡∣csc⁡(u)∣+c\int \cot(u) du = \ln|\sin(u)| + c = -\ln|\csc(u)| + c

    • ∫sec⁡(u)du=ln⁡∣sec⁡(u)+tan⁡(u)∣+c\int \sec(u) du = \ln|\sec(u)+\tan(u)| + c

    • ∫sec⁡3(u)du=12sec⁡(u)tan⁡(u)+ln⁡∣sec⁡(u)+tan⁡(u)∣+c\int \sec^3(u) du = \frac{1}{2}\sec(u)\tan(u) + \ln|\sec(u) + \tan(u)| + c

    • ∫csc⁡(u)du=ln⁡∣csc⁡(u)−cot⁡(u)∣+c\int \csc(u) du =\ln|\csc(u)-\cot(u)| + c

    • ∫csc⁡3(u)du=12(−csc⁡(u)cot⁡(u)+ln⁡∣csc⁡(u)−cot⁡(u)∣)+c\int \csc^3(u) du = \frac{1}{2}(-\csc(u)\cot(u) + \ln|\csc(u)-\cot(u)|) + c

    • Exponential & Logarithmic Functions

    • ∫eudu=eu+c\int e^u du = e^u + c

    • ∫audu=auln⁡(a)+c\int a^u du = a^u \ln(a) + c

    • ∫ln⁡(u)du=uln⁡(u)−u+c\int \ln(u) du = u \ln(u) - u + c

    • ∫eausin⁡(bu)du=eaua2+b2(asin⁡(bu)−bcos⁡(bu))+c\int e^{au}\sin(bu) du = \frac{e^{au}}{a^2+b^2}(a\sin(bu)-b\cos(bu)) + c

    • ∫eaucos⁡(bu)du=eaua2+b2(acos⁡(bu)+bsin⁡(bu))+c\int e^{au}\cos(bu) du = \frac{e^{au}}{a^2+b^2}(a\cos(bu)+b\sin(bu)) + c

    • ∫1uln⁡(u)du=ln⁡(ln⁡(u))+c\int \frac{1}{u}\ln(u) du = \ln(\ln(u)) + c

    • Inverse Trigonometric Functions

    • ∫1a2−u2du=sin⁡−1(ua)+c\int \frac{1}{\sqrt{a^2-u^2}} du = \sin^{-1}\left(\frac{u}{a}\right) + c

    • ∫sin⁡−1(u)du=usin⁡−1(u)+121−u2+c\int \sin^{-1}(u) du = u\sin^{-1}(u) + \frac{1}{2}\sqrt{1-u^2} + c

    • ∫1a2+u2du=1atan⁡−1(ua)+c\int \frac{1}{a^2+u^2} du = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + c

    • ∫tan⁡−1(u)du=utan⁡−1(u)−12ln⁡(1+u2)+c\int \tan^{-1}(u) du = u\tan^{-1}(u) - \frac{1}{2}\ln(1+u^2) + c

    • ∫1uu2−a2du=1asec⁡−1(ua)+c\int \frac{1}{u\sqrt{u^2-a^2}} du = \frac{1}{a} \sec^{-1}\left(\frac{u}{a}\right) + c

    • ∫cos⁡−1(u)du=ucos⁡−1(u)−1−u2+c\int \cos^{-1}(u) du = u\cos^{-1}(u) - \sqrt{1-u^2} + c

    • Hyperbolic Functions

    • ∫sinh⁡(u)du=cosh⁡(u)+c\int \sinh(u) du = \cosh(u) + c

    • \int \sech(u)\tanh(u) du = -\sech(u) + c

    • \int \sech^2(u) du = \tanh(u) + c

    • ∫cosh⁡(u)du=sinh⁡(u)+c\int \cosh(u) du = \sinh(u) + c

    • \int \csch(u)\coth(u) du = -\csch(u) + c

    • \int \csch^2(u) du = -\coth(u) + c

    • ∫tanh⁡(u)du=ln⁡∣cosh⁡(u)∣+c\int \tanh(u) du = \ln|\cosh(u)| + c

    • \int \sech(u) du = \tan^{-1}(\sinh(u)) + c

    • Miscellaneous

    • ∫1a2−u2du=12aln⁡∣u+au−a∣+c\int \frac{1}{a^2-u^2} du = \frac{1}{2a}\ln\left|\frac{u+a}{u-a}\right| + c

    • ∫a2+u2du=u2a2+u2+a22ln⁡∣u+u2+a2∣+c\int \sqrt{a^2+u^2} du = \frac{u}{2}\sqrt{a^2+u^2} + \frac{a^2}{2}\ln\left|u + \sqrt{u^2 + a^2}\right| + c

    • ∫1u2−a2du=12aln⁡∣u−au+a∣+c\int\frac{1}{u^2-a^2} du = \frac{1}{2a}\ln\left|\frac{u-a}{u+a}\right| + c

    • ∫u2−a2du=u2u2−a2−a22ln⁡∣u+u2−a2∣+c\int\sqrt{u^2-a^2} du = \frac{u}{2}\sqrt{u^2-a^2} - \frac{a^2}{2}\ln\left|u + \sqrt{u^2 - a^2}\right| + c

  • Standard Integration Techniques

    • u-Substitution:

    • ∫<em>abf(g(x))g′(x)dx=∫</em>g(b)g(a)f(u)du\int<em>a^b f(g(x)) g'(x)dx = \int</em>{g(b)}^{g(a)} f(u) du

    • Use u=g(x)u = g(x) and du=g′(x)dxdu = g'(x)dx. Drop limits for indefinite integrals.

    • Integration by Parts:

    • ∫udv=uv−∫vduand∫<em>abudv=uv∣</em>ab−∫abvdu\int u dv = uv - \int v du \quad \text{and} \quad \int<em>a^b u dv = uv\bigg|</em>a^b - \int_a^b v du

    • Choose uu and dvdv from the integral, compute dudu by differentiating uu, and compute vv using v=∫dvv = \int dv.

    • Trig Substitution:

    • For specific integrals involving roots:

      • ∫a2−b2x2<br>ightarrowx=absin⁡(θ)\int \sqrt{a^2-b^2x^2} <br>ightarrow x = \frac{a}{b} \sin(\theta)

      • ∫b2x2−a2<br>ightarrowx=absec⁡(θ)\int \sqrt{b^2x^2-a^2} <br>ightarrow x = \frac{a}{b} \sec(\theta)

      • ∫a2+b2x2<br>ightarrowx=abtan⁡(θ)\int \sqrt{a^2+b^2x^2} <br>ightarrow x = \frac{a}{b} \tan(\theta)

    • Partial Fractions:

    • For rational expressions ∫P(x)Q(x)dx\int \frac{P(x)}{Q(x)}dx where the degree of P(x)P(x) is less than Q(x)Q(x), factor the denominator completely and find the partial fraction decomposition.

    • Products and Quotients of Trig Functions:

    • For integrals like ∫sin⁡n(x)cos⁡m(x)dx\int \sin^n(x) \cos^m(x)dx:

      1. if nn is odd, strip one sine out and convert the rest to cosines.

      2. if mm is odd, strip one cosine out and convert the rest to sines.

      3. for both odd, use either method.

      4. if both are even, use double or half angle formulas to simplify the integral.

    • For ∫tan⁡n(x)sec⁡m(x)dx\int \tan^n(x) \sec^m(x)dx:

      1. if nn is odd, strip one tangent and one secant and convert the rest to secants.

      2. if mm is even, strip two secants and convert the rest to tangents.

      3. for both odd/even, choose method accordingly.

Example (Conversion): cos⁡6(x)=(cos⁡2(x))3=(1−sin⁡2(x))3\cos^6(x) = \left(\cos^2(x)\right)^3 = \left(1 - \sin^2(x)\right)^3

© Paul Dawkins - https://tutorial.math.lamar.edu