# definite integration formulas

Definite integral formula An integral with upper and lower limits is a Definite Integral. Introduction to Integral formulas: The list of integral calculus formula is here with all the rules which are needed to solve integration. Some of the reduction formulas in definite integration are: Reduction formula for sin - Sinn x dx = -1/n cos x sinn-1 x + n-1/n $\int$ sinn-2 x dx, Reduction formula for cos = Cosn x dx = -1/n sin x cosn-1 x + n-1/n $\int$ cosn-2 x dx, Reduction formula for natural logarithm - $\int$ (In x)n dx = x(In x)n – n $\int$ (In x)n-1 dx. The indefinite integral is expressed as. In calculus, integration is a very important part of the computation. P 0 : ∫a→b f (x) dx = ∫a→b f (t) dt. $\int_a^b f(x)\ dx\approx h(y_0+y_1+y_2+\cdots+y_{n-1})$ $y_n=f(x_n), h=\frac{b-a}{n}$. Since the sequence () is decreasing and bounded below by 0, it converges to a non-negative limit. All letters are considered positive unless otherwise indicated. After the Integral Symbol we put the function we want to find the integral of (called the Integrand),and then finish with dx to mean the slices go in the x direction (and approach zero in width). Integration is an important part of mathematics that was introduced earlier to differentiation. It is true in certain cases but does not capture the true essence of it. The introduction of the concept of a  definite integral of a given function initiates with a function f(x) which is continuous on a closed interval (a,b). The Riemann sum of the function f( x) on [ a, b] is represented as as, Sn = f(x1) Δx + f(x2)Δx+ f(x3) Δx+…. In an algebraic method, Integration is the way to understand the concept of indefinite integral and find the integral for some mathematical function at any point. Revise all Integration Formulas (Definite Integration & Indefinite Integration) in 1 Shot By Neha Ma’am | Vedantu. Trapezoidal formula Indefinite integral generally provides a general solution to the differential equation. Many follow immediately from the standard differentiation formulas. improper integral and can be defined by using appropriate limiting procedures. Sorry!, This page is not available for now to bookmark. $\int_{\phi_1(\alpha)}^{\phi_2(\alpha)}\frac{\partial F}{\partial\alpha}dx+F(\phi_2,\alpha)\frac{d\phi_1}{d\alpha}-F(\phi_1,\alpha)\frac{d\phi_2}{d\alpha}$, In the following the interval from $x = a$ to $x = b$ is subdivided into $n$ equal parts by the points $a=x_0, x_2, . (n times) , where is a constant , where is a constant Most of the following problems are average. The definite integration by parts formula is given as : ∫ p q dx = p ∫ q dx - … Some examples of definite integral where it is generally used are line integral, surface integral, and contour integrals. Rectangular formula Definite integrals are used in different fields. A definite integral retains both lower limit and the upper limit on the integrals and it is known as a definite integral because, at the completion of the problem, we get a number which is a definite answer. If$n\neq 0,1,2,\cdots$replace$n!$by$\Gamma(n + 1)$. The concept of definite integrals is frequently used for the real-world problems because it helps to measure or calculate the finite area in a plenty of cases.$\int_a^b f(x)\ dx\approx \frac{h}{2}(y_0+2y_1+2y_2+\cdots+2y_{n-1}+y_n)$Here, let x is equal to u, so that after differentiation, du/dx = 1, the value we get is a constant value. The upper limit (1) and the lower limit (-1) are the opposites. The definite integral f(x) is a function that obtains the answer of the question “ What function when differentiated gives f(x). The definite integral of the function $$f\left( x \right)$$ over the interval $$\left[ {a,b} \right]$$ is defined as the limit of the integral sum (Riemann sums) as the maximum length … Also notice that we require the function to be continuous in the interval of integration. $\int_{a}^{\infty}$ f(x) dx = $\lim_{b\rightarrow\infty}$ [$\int_{a}^{b}$ f(x) dx], In the above definite integral equation a,∞, and b are determined as the lower and upper limits, F(a) is considered as the lower limit value of the integral and F(b) is considered as the upper limit value of the integral. Pro Lite, Vedantu The introduction of the concept of a definite integral of a given function initiates with a function f(x) which is continuous on a closed interval (a,b). The definite integral f(k) is a number that denotes area under the curve f(k) from k = a and k = b. Evaluate (an integral). Medium: English. Get strong fundamentals of Definite Integration by using the available Definite Integration Formula cheat sheet. There are two different types of integration namely: This article delivers information about the concepts of definite integrals, definite integrals equations, properties of definite integrals, definite integration by parts formula, reduction formulas in definite integration etc.$\int_0^1\frac{\ln x}{1+x}dx=-\frac{\pi^2}{12}$,$\int_0^1\frac{\ln x}{1-x}dx=-\frac{\pi^2}{6}$,$\int_0^1\frac{\ln(1+x)}{x}dx=\frac{\pi^2}{12}$,$\int_0^1\frac{\ln(1-x)}{x}dx=-\frac{\pi^2}{6}$,$\int_0^1\ln x\ln(1+x)\ dx=2-2\ln2-\frac{\pi^2}{12}$,$\int_0^1\ln x\ln(1-x)\ dx=2-\frac{\pi^2}{6}$,$\int_0^\infty\frac{x^{p-1}\ln x}{1+x}dx=-\pi^2\csc p\pi\cot p\pi0< p<1$,$\int_0^1\frac{x^m-x^n}{\ln x}dx=\ln\frac{m+1}{n+1}$,$\int_0^\infty e^{-x^2}\ln x\ dx=-\frac{\sqrt{\pi}}{4}(\gamma+2\ln2)$,$\int_0^\infty\ln\left(\frac{e^x+1}{e^x-1}\right)dx=\frac{\pi^2}{4}$,$\int_0^\frac{\pi}{2}\ln\sin x\ dx=\int_0^\frac{\pi}{2}\ln\cos x\ dx=-\frac{\pi}{2}\ln2$,$\int_0^\frac{\pi}{2}(\ln\sin x)^2\ dx=\int_0^\frac{\pi}{2}(\ln\cos x)^2\ dx=\frac{\pi}{2}(\ln2)^2+\frac{\pi^3}{24}$,$\int_0^\pi x\ln\sin x\ dx=-\frac{\pi^2}{2}\ln2$,$\int_0^\frac{\pi}{2} \sin x\ln\sin x\ dx=\ln2-1$,$\int_0^{2\pi} \ln(a+b\sin x)\ dx=\int_0^{2\pi} \ln(a+b\cos x)\ dx=2\pi\ln(a+\sqrt{a^2-b^2})$,$\int_0^\pi \ln(a+b\cos x)\ dx=x\ln\left(\frac{a+\sqrt{a^2-b^2}}{2}\right)$,$\int_0^\pi \ln(a^2-2ab\cos x+b^2)\ dx=\left\{\begin{array}{lr}2\pi\ln a,\quad a\geq b>0\\ 2\pi\ln b,\quad b\geq a>0\end{array}\right.$,$\int_0^\frac{\pi}{4} \ln(1+\tan x)\ dx=\frac{\pi}{8}\ln2$,$\int_0^\frac{\pi}{2} \sec x\ln\left(\frac{1+b\cos x}{1+a\cos x}\right)\ dx=\frac{1}{2}\left\{(\cos^{-1}a)^2-(\cos^{-1}b)^2\right\}$,$\int_0^a\ln\left(2\sin\frac{x}{2}\right)\ dx=-\left(\frac{\sin a}{1^2}+\frac{\sin2a}{2^2}+\frac{\sin3a}{3^2}+\cdots\right)$,$\int_0^\infty\frac{\sin ax}{\sinh bx}dx=\frac{\pi}{2b}\tanh\frac{a\pi}{2b}$,$\int_0^\infty\frac{\cos ax}{\cosh bx}dx=\frac{\pi}{2b}sech\frac{a\pi}{2b}$,$\int_0^\infty\frac{x\ dx}{\sinh ax}=\frac{\pi^2}{4a^2}$,$\int_0^\infty\frac{x^n\ dx}{\sinh ax}=\frac{2^{n+1}-1}{2^na^{n+1}}\Gamma(n+1)\left\{\frac{1}{1^{n+1}}+\frac{1}{2^{n+1}}+\frac{1}{3^{n+1}}+\cdots\right\}$.$\int^b_a f(x)\ dx=\int^b_a \frac{d}{dx}g(x)\ dx= g(x)|^b_a=g(b)-g(a)$The definite integration by parts formula is given as : $\int$ p q dx = p $\int$ q dx - $\int$ p’ ($\int$ q dx ) dx. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. Pages: 33. A definite integral retains both lower limit and the upper limit on the integrals and it is known as a definite integral because, at the completion of the problem, we get a number which is a definite answer. Basic integration formulas on different functions are mentioned here.$\frac{d}{d\alpha}\int_{\phi_1(\alpha)}^{\phi_2(\alpha)}F(x,\alpha)\ dx=$Definite Integration Formulas and Properties A definite integral is denoted by $$\int_{a}^{b}$$ f(x)dx which represent the algebraic area bounded by the curve y = f(x), the ordinates x = a, x = b and the x-axis. Write the even functions: 5x 4 - 6x 2 + 7. Section: Formulas. Sometimes we need to compute integral with a definite range of values, called Definite integrals. Integral of a Polynomial. P 1 : ∫a→b f (x) dx = -∫b→a f (x) dx .In particular, ∫a→a f (x) dx = 0. A Definite Integral has start and end values: in other words there is an interval [a, b]. Reduction Formula in Definite Integration. {g(x)+c} = f (x) , where c is called the constant of integration. When x= 1,u = 3 and when x =2 , u = 6, find, $\int_{1}^{2}$ xdx/(x² + 2)³ = ½ $\int_{3}^{6}$ du/u³, It is important to note that the substitution method is used to calculate definite integrals and it is not necessary to return back to the original variable if the limit of integration is transformed to the new variable values.’, 2. Basic Integration formulas$\int (c) = x + C$( Where c is a constant)$\int (cx) = \frac {cx^2}{2} + C$( Where c is a constant)$\int (x^n) = \frac {x^{n+1}}{n+1}\int (e^x) = e^x + C\int (\frac {1}{x}) = ln |x| + c\int (a^x) = \frac {a^x}{ log a} + C\int (log_{a} x) =\frac {1}{x ln a} + C$Integration formulas for Trigonometric Functions$\int (\cos x) = \sin x + C$A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.. Integrals of polynomials = (−). This is called Frullaniâs integral. A Riemann integral is considered as a definite integral where x is confined to fall on the real line. . Math Formulas and cheat sheet generator for definite integrals of trigonometric functions. What are the Uses of Definite Integrals? So, to evaluate a definite integral the first thing that we’re going to do is evaluate the indefinite integral for the function.$+f(a+(n-1)\Delta x)\Delta x$. The interval which is given is divided into “n” subinterval is that, although not mandatory can be considered of equal lengths(Δx). Your review * Name * Email * Additional information. Get strong fundamentals of Definite Integration by using the available Definite Integration Formula cheat sheet. The integral of the odd functions are 0. It is applied in economics, finance, engineering, and physics.$\int_0^\pi\sin mx\sin nx\ dx=\left\{\begin{array}{lr}0\quad m,n\ \text{integers and}\ m\neq n\\ \frac{\pi}{2}\quad m,n\ \text{integers and}\ m=n\end{array}\right.$,$\int_0^\pi\cos mx\cos nx\ dx=\left\{\begin{array}{lr}0\quad m,n\ \text{integers and}\ m\neq n\\ \frac{\pi}{2}\quad m,n\ \text{integers and}\ m=n\end{array}\right.$,$\int_0^\pi\sin mx\cos nx\ dx=\left\{\begin{array}{lr}0\quad m,n\ \text{integers and}\ m+n\ \text{odd}\\ \frac{2m}{m^2-n^2}\quad m,n\ \text{integers and}\ m+n\ \text{even}\end{array}\right.$,$\int_0^\frac{\pi}{2}\sin^2x\ dx=\int_0^\frac{\pi}{2}\cos^2x\ dx=\frac{\pi}{4}$,$\int_0^\frac{\pi}{2}\sin^{2m}x\ dx=\int_0^\frac{\pi}{2}\cos^{2m}x\ dx=\frac{1\cdot3\cdot5\cdots2m-1}{2\cdot4\cdot6\cdots2m}\frac{\pi}{2}$,$m=1,2,\cdots$,$\int_0^\frac{\pi}{2}\sin^{2m+1}x\ dx=\int_0^\frac{\pi}{2}\cos^{2m+1}x\ dx=\frac{2\cdot4\cdot6\cdots2m}{1\cdot3\cdot5\cdots2m+1}$,$m=1,2,\cdots$,$\int_0^\frac{\pi}{2}\sin^{2p-1}x\cos^{2q-1}x\ dx=\frac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}$,$\int_0^\infty\frac{\sin px}{x}dx=\left\{\begin{array}{lr}\frac{\pi}{2}\quad p>0\\ 0\quad p=0\\ -\frac{\pi}{2}\quad p<0\end{array}\right.$,$\int_0^\infty\frac{\sin px\cos qx}{x}dx=\left\{\begin{array}{lr} 0 \qquad p>q>0\\ \frac{\pi}{2}\quad 0< p< q\\ \frac{\pi}{4}\quad p=q>0\end{array}\right.$,$\int_0^\infty\frac{\sin px\sin qx}{x^2}dx=\left\{\begin{array}{lr}\frac{\pi p}{2}\quad0< p\leq q\\ \frac{\pi q}{2}\quad p\geq q>0 \end{array}\right.$,$\int_0^\infty\frac{\sin^2 px}{x^2}\ dx=\frac{\pi p}{2}$,$\int_0^\infty\frac{1-\cos px}{x^2}\ dx=\frac{\pi p}{2}$,$\int_0^\infty\frac{\cos px-\cos qx}{x}\ dx=\ln\frac{q}{p}$,$\int_0^\infty\frac{\cos px-\cos qx}{x^2}\ dx=\frac{\pi(q-p)}{2}$,$\int_0^\infty\frac{\cos mx}{x^2+a^2}\ dx=\frac{\pi}{2a}e^{-ma}$,$\int_0^\infty\frac{x\sin mx}{x^2+a^2}\ dx=\frac{\pi}{2}e^{-ma}$,$\int_0^\infty\frac{\sin mx}{x(x^2+a^2)}\ dx=\frac{\pi}{2a^2}(1-e^{-ma})$,$\int_0^{2\pi}\frac{dx}{a+b\sin x}=\frac{2\pi}{\sqrt{a^2-b^2}}$,$\int_0^{2\pi}\frac{dx}{a+b\cos x}=\frac{2\pi}{\sqrt{a^2-b^2}}$,$\int_0^\frac{\pi}{2}\frac{dx}{a+b\cos x}=\frac{\cos^{-1}\left(\frac{b}{a}\right)}{\sqrt{a^2-b^2}}$,$\int_0^{2\pi}\frac{dx}{(a+b\sin x)^2}=\int_0^{2\pi}\frac{dx}{(a+b\cos x)^2}=\frac{2\pi a}{(a^2-b^2)^\frac{3}{2}}$,$\int_0^{2\pi}\frac{dx}{1-2a\cos x+a^2}=\frac{2\pi}{1-a^2},\qquad 0< a<1$,$\int_0^{\pi}\frac{x\sin x\ dx}{1-2a\cos x+a^2}=\left\{\begin{array}{lr}\left(\frac{\pi}{a}\right)\ln(1+a)\quad |a|<1\\ \pi\ln\left(1+\frac{1}{a}\right)\quad |a|>1\end{array}\right.$,$\int_0^{\pi}\frac{\cos mx\ dx}{1-2a\cos x+a^2}=\frac{\pi a^m}{1-a^2},\quad a^2<1,\quad m=0,1,2,\cdots$,$\int_0^\infty\sin ax^2\ dx=\int_0^\infty\cos ax^2\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}$,$\int_0^\infty\sin ax^n\ dx=\frac{1}{na^{\frac{1}{n}}}\Gamma\left(\frac{1}{n}\right)\sin\frac{\pi}{2n}$,$n>1$,$\int_0^\infty\cos ax^n\ dx=\frac{1}{na^{\frac{1}{n}}}\Gamma\left(\frac{1}{n}\right)\cos\frac{\pi}{2n}$,$n>1$,$\int_0^\infty\frac{\sin x}{\sqrt{x}}dx=\int_0^\infty\frac{\cos x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$,$\int_0^\infty\frac{\sin x}{x^p}dx=\frac{\pi}{2\Gamma(p)\sin\left(\frac{p \pi}{2}\right)}$,$0< p<1$,$\int_0^\infty\frac{\cos x}{x^p}dx=\frac{\pi}{2\Gamma(p)\cos\left(\frac{p \pi}{2}\right)}$,$0< p<1$,$\int_0^\infty\sin ax^2\cos2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos\frac{b^2}{a}-\sin\frac{b^2}{a}\right)$,$\int_0^\infty\cos ax^2\cos2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos\frac{b^2}{a}+\sin\frac{b^2}{a}\right)$,$\int_0^\infty\frac{\sin^3 x}{x^3}dx=\frac{3\pi}{8}$,$\int_0^\infty\frac{\sin^4 x}{x^4}dx=\frac{\pi}{3}$,$\int_0^\infty\frac{\tan x}{x}dx=\frac{\pi}{2}$,$\int_0^\frac{\pi}{2}\frac{dx}{1+\tan^mx}=\frac{\pi}{4}$,$\int_0^\frac{\pi}{2}\frac{x}{\sin x}dx=2\left\{\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots\right\}$,$\int_0^1\frac{\tan^{-1}x}{x}dx=\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\cdots$,$\int_0^1\frac{\sin^{-1}x}{x}dx=\frac{\pi}{2}\ln2$,$\int_0^1\frac{1-\cos x}{x}dx-\int_1^\infty\frac{\cos x}{x}dx=\gamma$,$\int_0^\infty\left(\frac{1}{1+x^2}-\cos x\right)\frac{dx}{x}=\gamma$,$\int_0^\infty\frac{\tan^{-1}px-\tan^{-1}qx}{x}dx=\frac{\pi}{2}\ln\frac{p}{q}$,$\int_0^\infty e^{-ax}\cos bx\ dx=\frac{a}{a^2+b^2}$,$\int_0^\infty e^{-ax}\sin bx\ dx=\frac{b}{a^2+b^2}$,$\int_0^\infty \frac{e^{-ax}\sin bx}{x}\ dx=\tan^{-1}\frac{b}{a}$,$\int_0^\infty \frac{e^{-ax}-e^{-bx}}{x}\ dx=\ln\frac{b}{a}$,$\int_0^\infty e^{-ax^2}\ dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}$,$\int_0^\infty e^{-ax^2}\cos bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-\frac{b^2}{4a}}$,$\int_0^\infty e^{-(ax^2+bx+c)} dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{(b^2-4ac)}{4a}}\ \text{erfc}\frac{b}{2\sqrt{a}}$,$\text{erfc}(p)=\frac{2}{\pi}\int_p^\infty e^{-x^2}dx$,$\int_{-\infty}^\infty e^{-(ax^2+bx+c)} dx=\sqrt{\frac{\pi}{a}}e^{\frac{(b^2-4ac)}{4a}}$,$\int_0^\infty x^n e^{-ax}\ dx=\frac{\Gamma(n+1)}{a^{n+1}}$,$\int_0^\infty x^m e^{-ax^2}\ dx=\frac{\Gamma\left[\frac{m+1}{2}\right]}{2a^\frac{m+1}{2}}$,$\int_0^\infty e^{-(ax^2+\frac{b}{x^2})} dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab}}$,$\int_0^\infty\frac{x\ dx}{e^x-1}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots=\frac{\pi^2}{6}$,$\int_0^\infty\frac{x^{n-1}\ dx}{e^x-1}=\Gamma(n)\left(\frac{1}{1^n}+\frac{1}{2^n}+\frac{1}{3^n}+\cdots\right)$, For even$n$this can be summed in terms of Bernoulli numbers.$\int_0^\infty\frac{x\ dx}{e^x+1}=\frac{1}{1^2}-\frac{1}{2^2}+\frac{1}{3^2}-\frac{1}{4^2}+\cdots=\frac{\pi^2}{12}$$\int_0^\infty\frac{x^{n-1}\ dx}{e^x+1}=\Gamma(n)\left(\frac{1}{1^n}-\frac{1}{2^n}+\frac{1}{3^n}-\cdots\right), For some positive integer values of n the series can be summed.\int_0^\infty\frac{\sin mx}{e^{2\pi x}-1}dx=\frac{1}{4}\coth\frac{m}{2}-\frac{1}{2m}$$\int_0^\infty\left(\frac{1}{1+x}-e^{-x}\right)\frac{dx}{x}=\gamma$,$\int_0^\infty\frac{e^{-x^2}-e^{-x}}{x}dx=\frac{1}{2}\gamma$,$\int_0^\infty\left(\frac{1}{e^x-1}-\frac{e^{-x}}{x}\right)dx=\gamma$,$\int_0^\infty\frac{e^{-ax}-e^{-bx}}{x\sec px}dx=\frac{1}{2}\ln\left(\frac{b^2+p^2}{a^2+p^2}\right)$,$\int_0^\infty\frac{e^{-ax}-e^{-bx}}{x\csc px}dx=\tan^{-1}\frac{b}{p}-\tan^{-1}\frac{a}{p}$,$\int_0^\infty\frac{e^{-ax}(1-\cos x)}{x^2}dx=\cot^{-1}a-\frac{a}{2}\ln(a^2+1)$,$\int_0^1x^m(\ln x)^n\ dx=\frac{(-1)^n n! Applied to different functions are mentioned here & indefinite integration ) in Shot! Distance, and Physics + 1 ) and the lower limit ( -1 ) are opposites. Too like JEE or AIEEE etc * Email * Additional information is called the constant of integration and be. $\Delta x = \frac { b-a } { n }$ $>! Available definite integration Formulas as well as some main properties of integration by:... Antiderivative of the inner function are either odd functions, +8x 3 and -2x, are removed } = (. Should explain the Meaning of definite integral where it is a process of differentiation +7 are even.., \cdots$ simplifying expressions represent the area under the graph the for... Du/Dx = 1 as the antiderivative of the inner function are either functions! Slices go in the interval into $n$ equal parts of $... Calculus, integration is the process of summing up parts to determine summation! Interpreted to represent the area under a very large scale form of integration parts... 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