What is Definite integral: Definition and 390 Discussions

In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus, and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others.
The integrals enumerated here are those termed definite integrals, which can be interpreted formally as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function. In this case, they are called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.

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  1. U

    Evaluate this definite integral

    Homework Statement $$ \displaystyle \int_0^{\infty} e^{-x} \dfrac{a\sin ax - \cos ax}{1+a^2} da $$ Homework Equations The Attempt at a Solution Evaluating this using integration by parts will be a cumbersome process and I don't even think that would give me the answer. Substitutions aren't...
  2. A

    Changing limits of integration - definite integral (without u sub)?

    Hello Can someone please tell me how is: \int_{-R}^{R} \frac{\cos mx}{x^2 + 1}\,dx = 2\int_{0}^R \frac{\cos mx}{x^2 + 1}\,dx where, m and R are positive real numbers This is how I'm trying to solve it... \int_{-R}^R \frac{\cos mx}{x^2 + 1}\,dx = \int_{-R}^0 \frac{\cos mx}{x^2 + 1}\,dx...
  3. D

    Evaluation of a definite integral

    I am looking at a solution to an question and I don't understand how the value of the following definite integral comes out to be zero ? The following function is evaluated from 0 to ∞ with r being the variable ## exp(-β^2r^2)r^nr^-1/(n-1)## That should read r raised to the power of...
  4. G

    Finding area given definite integral

    Question : https://www.physicsforums.com/attachments/71328 My question is how did the 2a and 2b come from?? Equations: Area of trapezoid =(a+b/2)(h) Attempt: I know that the area of a trapezoid is (a+b/2)(h) However why is there now a 2a and 2b in its place? Could it be related to the 2s...
  5. G

    Definite Integral Explained: Negative & Positive Areas

    Can anyone explain this to me? What does if mean that the area may sometimes be negative but that the area must be positive??
  6. C

    Trigonometric identity double definite integral

    Double integral of (52-x^2-y^2)^.5 2<_ x <_ 4 2<_ y <_ 6 I get up to this simplicity that results in a zero! 1-cos^2(@) - sin^2(@) = 0 This identity seems to be useless. HELP PLEASE.
  7. J

    How can you evaluate this integral using an analytic method?

    I know the value of the following definite integral \int_{a}^{b}ydx I also have a realtion x=f(y) i.e. x is an explicit function of y but I do not have y as an explicit function of x. The relation between x and y is generally non linear. Now I want to get the following definite...
  8. T

    Numerical integration methods applicable to a type of definite integral

    Numerical integration methods applicable to a type of definite integrl Hey, so I've been working on a program to numerically integrate an integral of the form ∫xnf(x) dx, LIM(0 to INF.) Here n can go to negative non integral values, say -3.7 etc. and f(x) is a function of sin, cos and...
  9. J

    Find indefinite integral function, if definite integral value is know

    Is this possible.. Say, a∫b f(x)dx = G(x)|x=b - G(x)|x=a = S, where S, a and b are known. Can we find G(x) ?
  10. J

    MHB Evaluate Definite Integral $(x-3)^2$ and $(x+4)^2$

    Evaluation of $\displaystyle \int_{-5}^{-7}\ln \left(x-3\right)^2dx+2\int_{0}^{1}\ln(x+4)^2dx$ My Try:: Let $(x-3) = t$ Then $dx = dt$ and changing Limit, we get and Again put $(x+4) = u,$ Then $dx = du$ and changing Limit, we get $\displaystyle...
  11. anemone

    MHB What is the Solution to This Definite Integral Challenge?

    Evaluate $\displaystyle\int^{\dfrac{\pi}{4}}_0 \dfrac{x}{(\sin x+\cos x)\cos x}\ dx$.
  12. J

    Solution for Tricky Definite Integral: How to Find I in Terms of A"

    Am trying to get a solution to the definite integral below. Looking for some direction. I = 0∫1 xf(x)dx where 0∫1 f(x)dx = A, is known. Also, its is know that when x =1, f(x) =0 and when x =0, f(x) =1. Can we get a solution of I in terms of A? I have tried going the...
  13. MarkFL

    MHB Solve Definite Integral w/ Absolute Value Factor - Yahoo! Answers

    ☺'s question at Yahoo! Answers: a definite integral whose integrand has an absolute value factor. Here is the question: I have posted a link there to this thread so the OP can view my work.
  14. Saitama

    MHB Limit with a definite integral

    Problem: Evaluate: $$\lim_{n\rightarrow \infty}\int_0^1 \frac{nx^{n-1}}{1+x}\,dx$$ Attempt: I used the series expansion: $$\frac{1}{1+x}=\sum_{r=0}^{\infty} (-1)^rx^r$$ From above, I got: $$\lim_{n\rightarrow \infty} \sum_{r=0}^{\infty} \frac{(-1)^rn}{n+r}$$ But I don't see how to proceed from...
  15. PhysicoRaj

    What Is the Best Approach to Integrate ln(sec θ - tan θ) from -π/4 to π/4?

    Homework Statement Integrate:I=\int_{-π/4}^{π/4} \ln{(\sec θ-\tan θ)}\,dθ Homework Equations Properties of definite integrals, basic integration formulae, trigonometric identities. The Attempt at a Solution By properties of definite integrals, the same integral I wrote as...
  16. I

    Solve Tricky Definite Integral with x^a-1 Over ln(x) on Interval 0 to 1

    I am trying to solve this integral: \int \frac{x^a-1}{ln(x)} dx (with the interval from 0 to 1). I have tried substitution but I could not find a way to get it to work. Any ideas on how to solve this? Thanks!
  17. Saitama

    MHB Evaluating a definite integral

    Problem: $$\int_0^{\infty} \frac{1}{x}\left(\frac{1}{1+e^x}-\frac{1}{1+e^{2x}}\right)\,dx$$ Attempt: I use the following two series expansions: $$\frac{1}{1+e^x}=\frac{e^{-x}}{1+e^{-x}}=e^{-x}\sum_{k=0}^{\infty} (-1)^k e^{-kx}=\sum_{k=0}^{\infty} (-1)^k e^{-(k+1)x}$$...
  18. J

    MHB Evaluation of definite Integral

    $\displaystyle \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 x}}dx = \int_{0}^{\frac{\pi}{4}}\frac{\pi}{2}-\int_{0}^{\frac{\pi}{4}}\cot^{-1}\sqrt{\frac{\cos 2x}{2\cos^2...
  19. A

    A Definite Integral Using the Residue Theorem

    Homework Statement I'm trying to solve this definite integral using the residue theorem: \int _0^\pi \frac{d \theta}{ (2+ \cos \theta)^2} Homework Equations I got the residue theorem which says that \oint_C f(z)dz = 2 \pi i \ \ \text{times the sum of the residues inside C}...
  20. MarkFL

    MHB Derivative of ∫ (1+v^3)^10 dv from sinx to cosx | Ashleigh N.

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  21. Saitama

    MHB Solving a definite integral without using gamma function

    Problem: Evaluate: $$\int_0^{\infty} t^{-1/4}e^{-t}\,dt$$ Attempt: I recognised this one as $\Gamma(3/4)$. I found a few formulas on Wolfram Mathworld website which helps to evaluate this but I am wondering if I can solve the definite integral from elementary methods (like by parts). Any help...
  22. Saitama

    MHB Evaluating a definite integral

    Problem: $$\int_1^e \frac{1+x^2\ln x}{x+x^2\ln x}\,\,dx$$ Attempt: I tried the substitution $\ln x=t \Rightarrow dx/x=dt$ and got the following integral: $$\int_0^1 \frac{1+e^{2t}t}{1+e^t t}\,dt$$ I am not sure how to proceed after this. :confused: Any help is appreciated. Thanks!
  23. Saitama

    MHB Definite Integral challenge #4

    Evaluate: $$2^{2009}\frac{\displaystyle \int_0^1 x^{1004}(1-x)^{1004}\,dx}{\displaystyle \int_0^1x^{1004}(1-x^{2010})^{1004}\,dx}$$ ...of course without the use of beta or gamma functions. :p
  24. MarkFL

    MHB Find Upper Limit of Integral: "Integration Please Help?

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  25. Saitama

    MHB Definite Integral challenge #3

    Evaluate the following: $$\int_0^{\pi} e^{\cos x} \cos(\sin x)\,\,dx$$
  26. Saitama

    MHB Minimizing a definite integral

    Problem: Find the value of $a$ such that $$\int_0^{\pi/2} |\cos(x)-ax^2|\,dx$$ is minimum. Attempt: Honestly, I don't know how to start. I tried the following: $$\int_0^{\pi/2} |\cos(x)-ax^2|\,dx \geq \int_0^{\pi/2}|\cos(x)|\,dx-\int_0^{\pi/2}|a|x^2\,dx=1-\frac{|a|\pi^3}{24}$$ $$\Rightarrow...
  27. Saitama

    MHB Integrating Symmetric Definite Integrals: A Trick for Evaluating $U_n-U_{n-1}$

    Attempt: If $\displaystyle U_n=\int_0^{\pi/2} \frac{\sin^2(nx)}{\sin^2x}\,dx$, then find $U_n-U_{n-1}$. Attempt: $$U_n=\int_0^{\pi/2} \frac{\sin^2(nx)}{\sin^2x}\,dx$$ $$U_{n-1}=\int_0^{\pi/2} \frac{\sin^2((n-1)x)}{\sin^2x}\,dx$$ $$\Rightarrow U_n-U_{n-1}=\int_0^{\pi/2}...
  28. Saitama

    MHB Evaluating a definite integral

    Problem: If f is continuous and differentiable function in $x \in (0,1)$ suuch that $\sum_{r=0}^{1}\left(f(x+r)-\left|e^x-r-1\right|\right)$=0, then $\int_0^{11} f(x)\,dx$ is A)65+4ln2-7e B)63+4ln2-9e C)69-9e D)29-23e Ans: A Attempt: I could only write the following...
  29. Saitama

    MHB Definite Integral challenge #2

    Evaluate: $$\Large \int_{\pi/2}^{5\pi/2} \frac{e^{\arctan(\sin x)}}{e^{\arctan(\sin x)}+e^{\arctan(\cos x)}}$$
  30. Saitama

    MHB Mathematical Techniques for Solving the Definite Integral Challenge

    Compute: $$\int_0^{\pi/2} \tan(x)\ln(\sin(x))\,dx$$
  31. JasonHathaway

    Definite integral approaches infinity

    Homework Statement 180\int_5^\propto \frac{2}{(4+x^{2})^{3/2}} dx Homework Equations Trigonometric Substitutions: (x=2 tan z). The Attempt at a Solution I've computed the integral and ended up with 180 [\frac{x}{2(4+x^2)^{1/2}}] from 5 to infinity. I could've easily computed...
  32. P

    MHB Definite Integral: Practice Problem Help

    My professor sent out an online work sheet with tons of practice problems, and I'm having trouble with this one, is my answer right? (see link) I chose this because a definite integral has to have limits, correct?
  33. M

    Definite Integral with Absolute Value.

    The problem is ∫x^2 - 3x - 5 with the lower limit being -4 and the upper limit 7. I broke the integrals into three parts from [-4, -1.1926], [-1.1926, 4.1926], [4.1926, 7] I did the integral and got (x^3)/3 - (3/2)x^2 - 5x I subbed in the lower and upper limits and got 32.861 for [-4...
  34. H

    Definite integral with x^2+c in the denominator

    Homework Statement Homework Equations solve the definite integral \int_{2.6}^{5.5} \frac{1}{x^2+9}dx The Attempt at a Solution ln(5.5^2+9)-ln(2.6^2+9) doesn't seem correct
  35. MarkFL

    MHB Calculate Definite Integral of arcos(tanx) from -pi/4 to pi/4

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  36. MarkFL

    MHB Optimize Definite Integral Function: Math Help | Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  37. L

    How Do You Solve the Integral of x^2 exp(-2amx^2/h)?

    I attached the solution from the solution manual of the integral I'm trying to figure out. \int_{-∞}^{∞}x^{2}exp(\frac{-2amx^{2}}{h}) The solution of that integral without the x2 in front is \sqrt{\frac{{\pi}h}{2am}} So with the x2 I assumed I needed to do integration by parts. So...
  38. A

    The most direct solution for a definite integral

    Hi, I'm wondering if I have the most direct solution for this integral or if there is a more efficient way of solving this. I haven't seen a double substitution deployed on one of these problems yet, so I thought perhaps this was not necessary. Homework Statement Using the substitution t...
  39. MarkFL

    MHB How to Compute a Definite Integral with Symmetry: The Case of $f(-x)=f(x)$

    Suppose $f(-x)=f(x)$, then compute the following definite integral: \int_{-a}^{a}\frac{1}{1+2^{f(x)}}\,dx where $0<a\in\mathbb{R}$.
  40. MarkFL

    MHB CALCULUS: Find Integral from -8 to -2 by Interpreting in Terms of Areas

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  41. MarkFL

    MHB Selena's question at Yahoo Answers regarding a definite integral by parts

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  42. S

    Definite Integral limit problems

    \int_0^{2\pi} \frac{1}{25cos^2(t) + 9sin^2(t)}dt Substituted the variables twice and got the upper and lower boundaries to both be 0 (think i might have gone wrong there) \frac{1}{15} tan^{-1} \frac{3tan(t)}{5} with upper and lower boundaries both being 0. I know the answer is 2\pi/15...
  43. Saitama

    Finding the Limit of a Definite Integral in an Integral Problem

    Homework Statement Let ##\displaystyle f(r)=\int_0^{\pi/2} x^r\sin x \,\, dx##. Now match the following List-I with List-II. $$ \begin{array} {|c| c | l c|} \hline & \text{List-I} & & \text{List-II} & & \\ \hline \text{(P)} & \lim_{r\rightarrow \infty}...
  44. I

    MHB Integration Question: Differentiating a definite integral

    So the question is…Evaluate the following… \frac{d}{dx} \left(\int _1^{x^2} \cos(t^2) \, dt \right) I thought i could use the FTC on this because it states… \frac{d}{dx} \left(\int_0^x f(t)\, dt \right)=f(x) but i can't correct? because in my question it starts at 1…is there some way to apply...
  45. I

    MHB Riemann Sum Definite Integral Question

    So the question is Evaluate (x-2)dx as the integral goes from -2 to 2 using the definition of a definite integral, choosing your sample points to be the right endpoints of the subintervals… Ok, so i understand how to do this problem if it gave me an actual number of interval like n=6 but it...
  46. J

    MHB Calculate Definite Integral of $\int_{0}^{1991}\{ \frac{2x+5}{x+1}\}[ x]dx$

    Calculation of : $\displaystyle \int_{0}^{1991}\{ \frac{2x+5}{x+1}\}[ x]dx$, where $[ x]$ and $\{ x \}$ denote the integral and fractional part of $x$ My Trial :: $\displaystyle \int_{0}^{1991}\left\{\frac{(2x+2)+3}{x+1}\right\}\cdot [x]dx$ $\displaystyle...
  47. A

    Definite Integral Homework: Equations and Solution Attempt

    Homework Statement Homework Equations The Attempt at a Solution
  48. Saitama

    Definite Integral Problem: Finding the Value of an Integral Using Substitution

    Homework Statement If the value of the integral ##\displaystyle \int_1^2 e^{x^2}\,\, dx## is ##\alpha##, then the value of ##\displaystyle \int_e^{e^4} \sqrt{\ln x} \,\, dx## is: A)##e^4-e-\alpha## B)##2e^4-e-\alpha## C)##2(e^4-e)-\alpha## D)##2e^4-1-\alpha## Homework Equations...
  49. DreamWeaver

    MHB Definite integral challenge....

    For m \in \mathbb{Z}^+, and a, \, z \in \mathbb{R} > 0, evaluate the definite integral:\int_0^z\frac{x^m}{(a+\log x)}\,dx[I'll be adding a few generalized forms like this in the logarithmic integrals thread, in Maths Notes, shortly... (Heidy) ]
  50. I

    Is There an Analytical Solution for This Tough Definite Integral?

    Hello, I would like to evaluate the following definite integral \int_0^1 \frac{exp(1/(x(1-x)))}{\sqrt{x(1-x)}} Numerically I get the result of about 1.4695 and it appears to converge nicely in the domain of interest (0;1). However, I'm wondering whether some kind of analytical integral...
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