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

Complex Numbers – HSC Math

A complex number is a number that can be expressed in the form $a + ib$, where a and b are real numbers, and $i$ is a solution of the equation $x^2 = −1$. Because no real number satisfies this equation, $i$ is called an imaginary number. Cartesian form $z=x+iy; x=r\ cos\ θ, t=r\ sin\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

Real Numbers – HSC Math

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. Real number set can be represented by the symbol R. If $a>b$ and $(x-a)(x-b)b$. So $b < x < a$. If $a>b$ and $(x-a)(x-b)>0$ then $x>a$ or $x<b$. So $x < b$ or $x >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

Application of Differentiation – HSC Math

if $(x,y) is a point over line $y=f(x)$ and the slope of the line at that point $(x,y)$ is θ$, then- $tan\ θ=\d/\dy(y)$ here, the tangent of the line at point $(x,y)$ creates θ° angle with $x\ axis$. If the tangent of a line is parallel to $x\ axis$ or perpendicular to $y\ axis$ then-Read More

Multiple Times Differentiation – HSC Math

if $y=f(x),$ then $\d/{\dx}(y)=\dy/{\dx}=f'(x)=y′=y_1$ $\d/{\dx}(y_1)=\d^2y/{\dx^2}=y_2$ $\d/{\dx}(y_2)=\d^3y/{\dx^3}=y_3$ $\d/{\dx}(y_{n-1})=\d^ny/{\dx^n}=y_n$ if $y=e^{mx}, y_n=m^n\ e^{mx}$ if $y=a^x, y_n=a^x\ (log_e\ a)^n=a^x\ (ln\ a)^n$ if $y=sin(ax+b), y_n=a^n\ sin(n.π/2+(ax+b))$ if $y=cox(ax+b), y_n=a^n\ cos(n.π/2+(ax+b))$

Differentiation Formulas – HSC Math

In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. if $y=f(x)$ is a function, then its first derivative, i.e. ${\dy}/{\dx}=f'(x)=\lim↙{h→0}{f(x+h)-f(x)}/h$ GeneralRead More

Functions – HSC Math

A function is a special relationship where each input has a single output. It is often written as $y=f(x)$ where $x$ is input and $y$ is the output value. Say $x_1,x_2∈D$ $f(x_1)=f(x_2)$ if $x_1=x_2$ or, $f(x_1)≠f(x_2)$ if $x_1≠x_2$ then the function is called a one-to-one function. Where D is the set of decimal numbers. ifRead More