Posts for Tag: Integration

Reduction Formulas

$∫x^ne^{mx}\ dx=1/mx^ne^{mx}-n/m∫x^{n-1}e^{mx}\ dx$ $∫{e^{mx}}/{x^n}\ dx=-e^{mx}/{(n-1)x^{n-1}}+m/{n-1}∫e^{mx}/{x^{n-1}}\ dx, n≠1$ $∫sin\ h^nx\ dx=1/nsin\ h^{n-1}x\ cos\ hx-{n-1}/n∫sin\ h^{n-2}x\ dx$ $∫{dx}/{sin\ h^nx}=-{cos\ hx}/{(n-1)sin\ h^{n-1}x}-{n-2}/{n-1}∫{dx}/{sin\ h^{n-2}x},n≠1$ $∫cos\ h^nx\ dx=1/nsin\ hx\ cos\ h^{n-1}x\ cos\ hx+{n-1}/n∫cos\ h^{n-2}x\ dx$ $∫{dx}/{cos\ h^nx}=-{sin\ hx}/{(n-1)cos\ h^{n-1}x}+{n-2}/{n-1}∫{dx}/{cos\ h^{n-2}x},n≠1$ $∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n+1}x\ cos\ h^{m-1}x}/{n+m}$ $+{m-1}/{n+m}∫sin\ h^n\ cos\ h^{m-2}x\ dx$ $∫sin\ h^nx\ cos\ h^mx\ dx={sin\ h^{n-1}x\ cos\Read More

Integral of Exponential and Logarithmic Functions

$∫e^x\ dx=e^x+c$ $∫a^x\ dx={a^x}/{ln\ a}+c$ $∫e^{ax}\ dx=e^{ax}/a+c$ $∫xe^{ax}\ dx=e^{ax}/a^2(ax-1)+c$ $∫ln\ x\ dx=x\ ln\ x-x+c$ $∫{dx}/{x\ ln\ x}=ln|ln\ x|+c$ $∫x^n\ ln\ x\ dx=x^{n+1}[{ln\ x}/{n+1}-1/(n+1)^2]+c$ $∫e^{ax}\ sin\ bx\ dx={a\ sin\ bx-b\ cos\ bx}/{a^2+b^2}e^{ax}+c$ $∫e^{ax}\ cos\ bx\ dx={a\ cos\ bx+b\ sin\ bx}/{a^2+b^2}e^{ax}+c$

Integrals of Hyperbolic Functions

$∫sin\ hx\ dx=cos\ hx+c$ $∫cos\ hx\ dx=sin\ hx+c$ $∫tan\ hx\ dx=ln\ cos\ hx+c$ $∫cot\ hx\ dx=ln|sin\ hx|+c$ $∫sec\ h^2x\ dx=tan\ hx+c$ $∫cosec\ h^2x\ dx=-cot\ hx+c$ $∫sec\ hx\ tan\ hx\ dx=-sec\ hx+c$ $∫cosec\ hx\ cot\ hx\ dx=-cosec\ hx+c$

Integrals of Trigonometric Functions

$∫sin\ x\ dx=-cos\ x+c$ $∫cos\ x\ dx=sin\ x+c$ $∫sin^2\ x \ dx=x/2-1/4sin\ 2x+c$ $∫cos^2\ x \ dx=x/2+1/4sin\ 2x+c$ $∫sin^3\ x \ dx=1/3cos^3\ x-cos\ x+c=1/12cos\ 3x-3/4cos\ x+c$ $∫cos^3\ x \ dx=sin\ x-1/3sin^3\ x+c=1/12sin\ 3x+3/4sin\ x+c$ $∫{dx}/{sin\ x}=∫cosec\ x\ dx=ln|tan{x/2}|+c$ $∫{dx}/{cos\ x}=∫sec\ x\ dx=ln|tan(π/4+x/2)|+c$ $∫{dx}/{sin^2\ x}=∫cosec^2\ x\ dx=-cot\ x+c$ $∫{dx}/{cos^2\ x}=∫sec^2\ x\ dx=tan\ x+c$ $∫{dx}/{sin^3\ x}=∫cosec^3\ x\Read More

Integrals of Irrational Functions

An irrational function is a function whose analytic expression has the independent variable $x$ under the root symbol. $∫{dx}/√{ax+b}=2/a√{ax+b}+c$ $∫√{ax+b}\ dx=2/{3a}(ax+b)^{3/2}+c$ $∫{xdx}/√{ax+b}={2(ax-2b)}/{3a^2}√{ax+b}+c$ $∫x√{ax+b}\ dx={2(3ax-2b)}/{15a^2}(ax+b)^{3/2}+c$ $∫{dx}/{(x+c)√{ax+b}}=1/√{b-ac}\ ln|{√{ax+b}-√{b-ac}}/{√{ax+b}+√{b-ac}}|+c_1$, $b-ac>0$. $∫{dx}/{(x+c)√{ax+b}}=1/√{ac-b}\ tan^{-1}{√{ax+b}/√{ac-b}}+c_1$, $b-ac0$ $∫√{{ax+b}/{cx+d}}\ dx$ $=1/c√{(ax+c)(cx+d)}-{ad-bc}/{c√{ac}}\ tan^{-1}{√{{a(cx+d)}/{c(ax+b)}}+c_1,(a0)$ $∫x^2√{a+bx}\ dx={2(8a^2-12abx+15b^2x^2)}/{105b^3}√(a+bx)^3+c$ $∫x^2/√{a+bx}\ dx={2(8a^2-4abx+3b^2x^2)}/{15b^3}√(a+bx)+c$ $∫{dx}/{x√{a+bx}}=1/√a\ ln|{√{a+bx}-√a}/{√{a+bx}+√a}|+c, a>0$ $∫{dx}/{x√{a+bx}}=2/√{-a}\ tan^{-1}({a+bx}/{-a})+c, a0$ $∫{dx}/√{ax^2+bx+c}=-1/√a\ sin^{-1}({2ax+b}/{4a}√{b^2-4ac})+c, aa$ $∫{dx}/{(x+b)√{a^2-x^2}}=1/√{a^2-b^2}\ ln|{x+b}/{√{a^2-b^2}√{a^2-x^2}+a^2}|+c, b<a$ $∫{dx}/{x^2√{a^2-x^2}}=-√{a^2-x^2}/{a^2x}+c$ $∫(a^2-x^2)^{3/2}\ dx=x/8(5a^2-2x^2)√{a^2-x^2}+{3a^4}/8\ sin^{-1}x/a+c$ $∫(a^2-x^2)^{-3/2}\Read More

Integrals of Rational Functions

$∫a\ dx=ax+c$ $∫x\ dx=x^2/2+c$ $∫x^2\ dx=x^3/3+c$ $∫x^n\ dx=x^{n+1}/{n+1}+c, n≠-1$ $∫(ax+b)^n\ dx=(ax+b)^{n+1}/{a(n+1)}+c, n≠-1$ $∫{dx}/x=ln\ |x|+c$ $∫{dx}/(ax+b)=1/a\ ln\ |ax+b|+c$ $∫{ax+b}/{cx+d}\ dx=a/c\ x+{bc-ad}/{c^2}ln\ |cx+d|+c$ $∫{dx}/{(x+a)(x+b)}=1/{a-b}ln\ |{x+b}/{x+a}|+c, a≠b$ $∫{x\ dx}/{a+bx}=1/{b^2}\ (a+bx-a\ ln|a+bx|)+c$ $∫{x^2\ dx}/{a+bx}=1/{b^3}\ [1/2(a+bx)^2-2a(a+bx)+a^2\ ln|a+bx|]+c$ $∫{dx}/{x(a+bx)}=1/a\ ln|{a+bx}/x|+c$ $∫{dx}/{x^2(a+bx)}=-1/{ax}+b/a^2\ ln|{a+bx}/x|+c$ $∫{x\ dx}/(a+bx)^2=1/b^2(ln\ |a+bx|+a/{a+bx})+c$ $∫{x^2\ dx}/(a+bx)^2=1/b^3(a+bx-2a\ ln\ |a+bx|-a^2/{a+bx})+c$ $∫{dx}/{x(a+bx)^2}=1/{a(a+bx)}+1/a^2\ ln|{a+bx}/x|+c$ $∫{dx}/{x^2-1}=1/2ln\ |{x-1}/{x+1}|+c$ $∫{dx}/{1-x^2}=1/2ln\ |{1+x}/{1-x}|+c$ $∫{dx}/{a^2-x^2}=1/{2a}ln\ |{a+x}/{a-x}|+c$ $∫{dx}/{x^2-a^2}=1/{2a}ln\ |{x-a}/{x+a}|+c$ $∫{dx}/{1+x^2}=tan^{-1}x+c$ $∫{dx}/{a^2+x^2}=1/a\Read More

Indefinite Integral Formulas

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Functions: $f,g,u,v$ Independent variables: $x,t,ξ$ Indefinite integral of a function: $∫f(x)dx,∫g(x)dx$ Derivative of a function: $y'(x), f'(x),F'(x)$ Real constants: a, b, c, d, k Natural numbers: m, n, i,Read More

Integration Formulas – HSC Math

$∫1/\x\ \dx=ln|x|+c$ $∫e^{mx}\ \dx=1/m\ e^{mx}+c$ $∫a^xd^x\ \dx=a^x/{ln\ a}+c, a>0, a≠1$ $∫cos\ x\ \dx=sin\ x+c$ $∫sin\ x\ \dx=-cos\ x+c$ $∫sec^2\ x\ \dx=tan\ x+c$ $∫cosec^2\ x\ \dx=-cot\ x+c$ $∫sec\ x\ tan\ x\ \dx=sec\ x+c$ $∫cosec\ x\ cot\ x\ \dx=-cosec\ x+c$ $∫tan\ x\ \dx=-ln|cos\ x|+c=ln|sec\ x|+c$ $∫cot\ x\ \dx=ln|sin\ x|+c$ $∫sec\ x\ \dx=ln|tan(π/4+x/2)|+c$ $∫sec\ x\ \dx=ln|sec\ x+tan\ x|+c$Read More