What is Indefinite integral: Definition and 207 Discussions

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G.Antiderivatives are related to definite integrals through the fundamental theorem of calculus: the definite integral of a function over an interval is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval.
In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between position, velocity and acceleration). The discrete equivalent of the notion of antiderivative is antidifference.

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  1. R

    Indefinite Integral - By parts works right?

    Nevermind, it's late and I realized why it doesn't work because I forgot to take into consideration that the denominator is (1/polynomial) Anyone care to explain to me how to do it the proper way? 1. Question 1 \int (x+2)/(x²+x+1) dx The only reason I ask is because my teacher...
  2. H

    Indefinite integral and anti-derivative

    Homework Statement Find the indefinite integral of 16x^2+36+1/(16x^2+36) with respect to x Homework Equations Anything possible to take an anti-derivative The Attempt at a Solution I have absolutely no idea on how to deal with this problem. I can take an anti-derivative of the...
  3. James889

    Integrating tan x ln x cos x to Solving Indefinite Integrals

    Hi, Looking to integrate the indefinite integral: \int tan~x\cdot ln~x\cdot cos~x Since tan x = sin x/ cos x, this integral be written as \int sin~x\cdot ln~x In that case i thought the answer was cot x. But that is wrong. Do you need to use integration by parts on this one?
  4. C

    How to Solve the Indefinite Integral with Trigonometric Substitution?

    Homework Statement \int{ x^3 \sqrt{(36-x^2)}dx} Homework Equations The Attempt at a Solution I tried using trig substitution but got 7776\int{cos^3(\theta)-cos^5(\theta)d\theta} which seems completely wrong 6cos(\theta)=x 6sin(\theta)d\theta=dx...
  5. M

    How to Find Indefinite Integral Using U-substitution

    Using U-substitution find the indefinite integral of: [sin(2x)/cos^4(2x)] dx So I do know that it will have to come out to it being ln... here's what i did so far ok so i made u= cos^4(2x) du= -8cos^3(2x)*sin(2x)dx...(just took the derivative of u and simplified it) so made sin(2x)dx=...
  6. P

    Indefinite Integral: Justification for Dropping Absolute Value Bars

    In a book while doing an indefinite integral the author first wrote (sec2 x)1/2 = |sec x| , fine , then the author says the following : "since we are doing an indefinite integral we can drop the absolute value bars" , now what is the justification for this ?
  7. W

    Evaluating an indefinite integral

    Homework Statement Evaluate the indefinite integral. \int \left({\sqrt[5]{x^5}}-\frac{6}{5 x}+\frac{1}{4 x^{7}} \right) dx The Attempt at a Solution O.k. the only anti-derivative I am having trouble getting is the first one {\sqrt[5]{x^5}}. I am not sure what formula I would use...
  8. L

    Tricky Logarithmic Indefinite Integral

    Homework Statement Hi, Our professor has only taught us these methods for Integration...thus far: Direct Integration Substituion Method So theoretically we should be able to solve this problem without using Integration by parts or partial fractions...: Integrate...
  9. N

    Solving Indefinite Integral: Approach and Techniques

    Homework Statement Solve the indefinite integral Homework Equations \int\frac{dy}{y(1-y)} How do I best approach this problem? I have been stuck for hours!
  10. L

    Indefinite Integral of (1/x^2)

    Homework Statement Hello, first of all I would like to apologize for the fact that this question is extremely trivial compared to the other questions being asked. I have a improper integral problem, and the entire problem itself is not relevant, because I understand everything in it except...
  11. S

    Integral: ∫(exp(6x))/(exp(12x)+25)dx Solution

    Homework Statement ∫(exp(6x))/(exp(12x)+25)dx Homework Equations answer: -arctan[5/exp(6*x)]/30 The Attempt at a Solution honestly, don't know where to start. i was looking at another forum and tried to set u=exp(x) du=exp(x) and dx=du/u. plugging that in i got u^6/(u^12+25)*du/u. not sure...
  12. 4

    Solving for indefinite integral

    Homework Statement My daughter at college asked me to help her with these but it's been years since I've done them. I said I would try and then look over what she comes up with so any help would be great not so I can give her the answers but so I can tell her whether or not she on the right...
  13. E

    Indefinite integral and average distance

    Homework Statement Find the average distance to the x-axis for points in the region bounded by the x-axis and the graph of y = x - x^2. Homework Equations The Attempt at a Solution Can someone guide me how to solve this?
  14. N

    Integrating Complexity: Indefinite Integral of e^(4x+(e^4x))

    Homework Statement Indefinite integral: e^(4x+(e^4x)) Homework Equations I'm thinking integration by parts, involving UV minus integral of Vdu The Attempt at a Solution So I saw that this can be split into two: e^(4x) times e^(e^4x)). The latter is a bit complicated. I...
  15. W

    Indefinite integral of vector function

    Homework Statement If {\vec{V}(t) is a vector function of t , find the indefinite integral: \int (\vec{V}\times \frac{d^2\vec{V}}{dt^2}) \,dt Homework Equations The Attempt at a Solution I have solved it by decomposing and integrating each terms of vector \vec{V}\times \frac{d^2t}{dt^2}...
  16. C

    Find the general indefinite integral

    Homework Statement I wasn't really sure where to post this, as I don't need help understanding the integration. I need help with the trigonometry! That being said, here is the problem. Find the general indefinite integral of \int\frac{\sin{x}}{1-sin^2{x}}dx Homework Equations...
  17. M

    How Do You Solve the Integral of (sec^2(sqrt(x)))/sqrt(x) Using u-Substitution?

    Find \displaystyle\int\dfrac{\sec ^2\sqrt{x}}{\sqrt{x}} dx We're supposed to use u du substitution but I can't seem to get this one. EDIT: Sorry I didn't read rules. I tried u=\sec^2\sqrt{x} and all variants. Usually it was in the form of [sec or cos][^2 or none][sqrt x]
  18. F

    Can You Solve this Tough Indefinite Integral?

    A long time since i posted at physics forums. Anyways, try helping me solve the following integral \int\frac{1}{x^{2n} + 1}dx I tried many ways but all futile. The best way with which i could come up was factorising the denominator by de moivre's theorem. By finding the 2nth roots of...
  19. G

    Solving an indefinite integral

    Homework Statement 1. The calculation of the probability of excitation of an atom originally in the ground state to an excited state, involves the contour integral INTEGRAL(-INF TO +INF) [S exp(iwt)/(t2 + s2)2dt Evaluate the above integral. Homework Equations The Attempt at a...
  20. F

    Evaluation of Indefinite Integral

    Homework Statement Evaluate the following indefinite integral: (2t6-3)/t3 dt The Attempt at a Solution I know I need to substitute. Tried u= t3 and found du= 3t2dt. Tried to find where du would substitute in, but found...
  21. P

    Getting a delta function from an indefinite integral

    Hey everybody, One question that I've had for a week or so now is how the following integral can equal a Dirac delta function: \frac{1}{2\pi} \int_{-\infty}^{\infty}{dt} \:e^{i(\omega - \omega^{'})t}\: = \: \delta(\omega - \omega^{'}) A text that I was reading discusses Fourier transforms...
  22. F

    Indefinite Integral of an Absolute Convergent Function

    Hi, I was wondering if a function is absolutely convergent over a certain interval, say, (0,\infty) will its indefinite integral also be absolutely convergent over the same interval? Also, assume that f(x) is convergent for (0,\infty). Would g(x) = \int{\int_{0}^{\infty}f(x)dx}dy &=&...
  23. E

    Evaluate this indefinite integral

    Evaluate this indefinite integreal S = integral S 1/(9+x^2)^2 This has been driving me and my friend nuts. We tried partial fractions only to realize that it brings us back to the same thing because its not a polynomial over a polynomial, we tried by parts and it did not help and we...
  24. M

    Indefinite Integral of e raised to a negative fraction

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  25. A

    What's the indefinite integral formula?

    Homework Statement evaluate the integral ∫x^2 sinpi x dx Homework Equations ∫u dv= uv - ∫v du integration by parts formula The Attempt at a Solution u=x^2 dv= sin pi x dx du = 2x v = -cos pi x dx ? the pi is giving me trouble
  26. A

    What's the indefinite integral?

    of sin pi x dx i thought it would be - cos pi x dx but i think it might be (1/pi) -cos pi x dx
  27. M

    How do I correctly solve the indefinite integral of [sin^3*(13x)*cos^8*(13x)]dx?

    The integral of [sin^3*(13x)*cos^8*(13x)]dx I think u=sin^3 so du/dx=cos^3 du=cos^3dx but then I am really not sure if that is correct, the trig functions confuse me a bit. Please help me, thank you!
  28. M

    What is the Indefinite Integral of [(e^(4x))/(e^(8x))+9]dx?

    Evaluate the indefinite integral: [(e^(4x))/(e^(8x))+9]dx -I think that u=e^(2x) so then du=e^(2x)dx then the answer would end up being [(e^(4x)+9)/(-1)]^(-1) but it was incorrect; I think that my u might be wrong and that's where the problem is, but I am not sure. Please help, thank...
  29. R

    Indefinite Integral of 5*(sin(6x)/sin(3x)) - Help Needed

    Homework Statement The integral of 5*(sin(6x)/sin(3x))dx The Attempt at a Solution I'm not quite sure what to do with this one. I moved 5 to the left of the integral, but then I'm lost. Apparently I'm rusty on these trig identities. Could anyone help me get started? Thank you.
  30. P

    Indefinite integral and Fundalmental of calculus?

    Homework Statement What is the connection between the Indefinite integral and the Fundalmental theorem of calculus (1st part)? The Attempt at a Solution They are the same to me but the FT is more formal.
  31. U

    Mastering Indefinite Integrals: Tips and Tricks for Evaluating Tricky Functions

    How can I evaluate \int\frac{dx}{\sqrt{1-2x-x^2}} using inverse trig functions? Thanks.
  32. U

    Another indefinite integral question?

    How to evaluate \int{\frac{\arccos{x}}{x^2}\,dx}?
  33. U

    Mastering Integration by Parts: Proving the Indefinite Integral Formula

    Hi, I need help evaluating the following integral by integration by parts: \int(a^2-x^2)^n\,dx. Specifically I am supposed to prove the following formula: \int(a^2-x^2)^n\,dx=\frac{x(a^2-x^2)^n}{2n+1}+\frac{2a^2n}{2n+1}\int(a^2-x^2)^{n-1}\,dx+C Any hints would be appreciated. Also, does...
  34. Y

    How to evaluate an indefinite integral

    how to evaluate the indefinite integral \int \frac{1}{\sqrt{x^2-1}} dx
  35. Y

    How to Evaluate Indefinite Integrals with Radical Expressions?

    how to evaluate the indefinite integral ∫(x^2-1)^(-1/2) dx and ∫x^(-1)*(1-x^2)^(-1/2) dx
  36. 0

    Don't understand why an indefinite integral is valid only on a interval

    Don't understand why "an indefinite integral is valid only on a interval" Hi I'm using Stewart's Calculus, in the section of indefinite integral, they say: "Recall from Theorem 4.10.1 that the most general antiderivative on a given interval is obtained by adding a constant to a particular...
  37. E

    Solve Int. 1: Find x^7, x^5, x^3 Terms in Answer

    1. \int(x^{2} + 5)^{3}dx This is what the book gives as the answer 1/7x^{7} + 3x^{5} + 25x^{3} + 125x + C I got something way different. Where are they getting the 3x^5 and 25x^3 from? Thanks. -v.b.
  38. R

    Solving Indefinite Integral: F(x)= \int\frac{1}{t}dt from x to 2x

    Homework Statement How is this function continuous from 0 to infinity F(x) = \int\frac{1}{t}dt from x to 2x Homework Equations I am fairly sure that this equation uses the properties of natural logs to solve. Also an infinite function has a derivative that is equal to 0. The...
  39. A

    Complete ALGO to solve a indefinite integral ( classroom questions )

    Helo everyone, can somebody post the best algorithm/strategy to solve indefinite integral questions which are usually asked to undergraduates. The most general set of steps that can be applied to every question one encounters in the classroom. Algo that though may be proved to be inconvenient...
  40. A

    Integrate x^3/(x^5-1): Solutions

    Hi, here is the question integrate x^3/(x^5-1)
  41. O

    I on this indefinite integral

    Homework Statement ok I am given this problem indef. int (1+tan^2*5x)dx i need to use the u subsitution method to find the answer but i cannot seem to find what to subsitute the worksheet says the answer is " one-fifth*tan5x+C Homework Equations The Attempt at a Solution
  42. C

    How Do I Solve the Integral of cos(x)sin(x)dx?

    Problem: [int]cosx(sinx)dx Given: x=pi; f(pi)=13.4 I am utterly confused on how to solve this integral. I am 99% positive (which is nothing in the math world) that I need to apply the product rule to all of this in order to find the antiderivative. However, no matter how I think of going about...
  43. I

    Can the Chain Rule Be Applied in Reverse for Integrating Functions Like u^n?

    There is a simple formula for calculating \frac{df(x)}{dx} u^n where u is a function of x and n is a positive rational number: \frac{df(x)}{dx} u^n = nu^{n-1} \ast \frac{du}{dx} . Is there a similar formula for calculating \int u^n dx where u is a function of x and n is a positive rational...
  44. S

    Evaluating an Indefinite Integral of cos^4(x)sin(x)dx

    evaluate the indefinite integral cos^4(x)sin(x)dx I tried using the half angle formula but this gives me a much more difficult integral, so i resorted to just regular substitution but am not sure if I can do this. u = cos(x) du = -sin(x)dx indefinite integral -u^4du then -1/5(u)^5...
  45. G

    How Do You Integrate (e^ax)cos^2(2bx)dx Correctly?

    I'm supposed to integrate the following expression, and supposedly there is a very simple way to do so. Maple comes up with something rediculous, so I'd appreciate any input. Sorry about the short hand, don't know how to make everything pretty on here: Integral[(e^ax)cos^2(2bx)dx] where a and...
  46. S

    Indefinite integral substitution

    evaluate the indefinite integral ((e^x)/((e^x)+1))dx I let u = ((e^x)+1) then du = (e^x)dx which occurs in the original equation so.. indefinite ingegral ((u^-1)du) taking the antiderivative I get 1 + C is this right? thanks!
  47. C

    Indefinite integral of sin^4xdx

    Hi everyone. I'm having some trouble evaluating the following integral \int{sin^4xdx} First let me start off by showing what I did. = \int{(sin^2x)(sin^2x)dx} =\int{[\frac{1}{2}-\frac{1}{2}cos(2x)] \ [\frac{1}{2}-\frac{1}{2}cos(2x)]dx...
  48. N

    Integrate this indefinite integral: 1/(x-6)^2 dx

    I need to integrate this indefinite integral: 1/(x-6)^2 dx Here is my work... Let u= x-6 du/dx=1 so: integral 1/u^2 du = 3/u^3 + c (constant) =3/(x-6)^3 + c Have I gone wrong? And if so where? Thanks
  49. A

    Solving Indefinite Integrals: "int (1/(sqrt -x^2 -2x))dx

    I have: int (1/(sqrt -x^2 -2x))dx so I rewrite (-x^2-2x) --> 1-(x+1)^2 and swap those two. then I say t=x+1, and substitute that in. So now I have: int (1/(sqrt 1-t^2)) dt Here I get stuck, can anyone please help?
  50. D

    Evaluating indefinite integral - toughie

    Evaluating indefinite integral -- toughie! I have the velocity function v(x) = [(k*x^2)/(2*m)] + v0 I need to integrate this to get position as a function of time. So v = dx/dt. Separating variables, I get t = Integral [2m/(2mv0 + kx^2)] Here's where I'm stuck...If i pull out the 2m, then I...
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