What is Antiderivative: Definition and 177 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. S

    How to find the antiderivative of cot(x)

    5) As you can see ln(sin(1)) +1/ln.5 does not yield one of the above answers. Can someone tell me where I went wrong?
  2. K

    Antiderivative of a Complex Exponential Derivative

    Homework Statement Find the antiderivative of each derivative: dp/dx = [(e^x)+(e^-x)]/[((e^x)-(e^-x))^2] The Attempt at a Solution I can't entirely figure out how to get myself started here. I tired splitting up the derivative so that it's [(e^x)/ (((e^x)-(e^-x))^2)] +...
  3. S

    Antiderivative of product of trig functions

    Homework Statement Find the antiderivative of sin^4(x)tan^2(x) Homework Equations Trig identities I may have overlooked. The Attempt at a Solution I tried writing the integrand in terms of sin and cos but that didn't seem to lead anywhere. I tried integration by parts since the...
  4. H

    Partial fractions to determine antiderivative of sec x

    Homework Statement Derive a formula for the antiderivative of sec x using the identity that sec x= cos x/ (1-sin^2x). Use a substitution for sin x and then partial fractions. Then multiply the solution by (1+sin x)/ (1+sin x) to obtain the more familiar formula for the antiderivative...
  5. B

    Finding the Antiderivative of 2xy

    Homework Statement It is simple: find the antiderivative of 2xy. Homework Equations The Attempt at a Solution I am inclined to say that it equals (xy)^2 +c, but can't help but feel i have left out something.
  6. S

    Possible to take antiderivative of (sin(x)/e^x)dx?

    Homework Statement There is an integral, according to an online integral calculator, but I cannot obtain it by integration by parts. Is there a way to obtain it without it being given to you? The derivative in question is [sin(x)/e^x]dx Homework Equations The Attempt at a Solution
  7. S

    Antiderivative: natural logs, exponents

    Homework Statement I am trying to anti-differentiate (3t+2e^t)*[e^(-3t/2)] Homework Equations The Attempt at a Solution I have simplified this into several forms, including [3t(e^(-3t/2)+2e^(-t/2)] . This form may or may not be accurate, but I am pretty sure that the form given...
  8. A

    How Do You Find the Antiderivative of \( \frac{2}{2x+1} \)?

    2/2x+1 how do i figure this out
  9. B

    Find constants and antiderivative problem

    Homework Statement Find constants c1 and c2 such that F(x) = c1xsinx+c2cosx is an antiderivative of f(x) = 4xcos(x)+3sin(x) Homework Equations d/dx cosx = -sinx d/dx sinx = cosx The Attempt at a Solution Don't know how to setup or handle a problem such as this. Prolly much simplier...
  10. L

    How do I find the antiderivative of -tanx on the interval (-π/2, π/2)?

    Hi, I'm just starting to learn integration (I went to the first class in my summer course yesterday), and I'm already a bit confused. I was doing some practice problems in the text, and I found this: Find the most general antiderivative of f on the indicated interval: f(x) = -\tan x on...
  11. G

    Antiderivative to the formula f(x)=(2xr-x²)^(1/2)

    Hi, i can't seem to get the antiderivative to the formula f(x)=(2xr-x²)^(1/2) right. Can anyone help me out? By the by, if could anyone link a site with formulas for antiderivatives that would be nice.
  12. B

    An integral with a fraction in it- finding the antiderivative

    An integral with a fraction in it--need help finding the antiderivative Homework Statement I have the integral \int(\frac{20}{1+x^2} - 2)dx where x = ± 3. To solve, I need to find the antiderivative of \frac{20}{1+x^2} - 2 Now, the worksheet actually gives the answer to the problem: The...
  13. C

    Antiderivative of arctan and x function (by parts maybe? )

    antiderivative of arctan and x function (by parts... maybe??!) find antiderivative of 2xarctan(8x) { = antiderivative s thing. I did it by parts at first and got x^2arctan(8x) - 8 * { (x^2)/(1+64x^2) To get the antiderivative of the second part i did it by parts again and that...
  14. C

    What is the antiderivative of (1-cos(t/6))^3 * sin(t/6)?

    Hey guys! So i have to find the antiderivative of (1-cos(t/6))^3 * sin(t/6) I have tried a bunch of different ways like by parts, trig substitution, identities, u-substitution then developing it, etc. but i just can't get it! Can some one just tell me the proper method to use after that I am...
  15. R

    What is the Antiderivative of x^(-2) with Given Conditions?

    Find f. (x>0) f''(x) = x^(-2) x > 0 f(1) = 0 f(8) = 0 Alright, everything was fine until one point...okay. First derivative: f''(x)=x^{-2} f'(x)=-x^{-1}+C Now here's where I'm not sure: in the next step, it will be x raised to zero, which is one. times 1/0, which is zero...
  16. V

    Antiderivative and Integral Calculus: Solving for the Correct Answer

    Homework Statement \int (X-1) \sqrt{X} dx The Attempt at a Solution It is multiple choice, I believe the answer is either (2/5)x^(5/2)-(2-3)x^(3/2)+c or (1/2)x^2+2x^(3/2)-x+c I have tried to find the derivatives of both these answers yet neither of them gave me the correct...
  17. L

    Finding the Anti-Derivative of (sin x)^3/2 - Tips and Tricks

    [SOLVED] Antiderivative of (sin x)^3/2 How do I find the anti-derivative of: \int sin^{3/2}x I can't see where to start... Thanks
  18. E

    Finding an Antiderivative in an Annulus

    [SOLVED] Finding an Antiderivative in an Annulus Homework Statement Let A be the annulus, A = {z : 0 <= r < |z| < R}. Suppose that g is an analytic function on A (and that g' is continuous) with the property that \int_{|z| = s} g(z) \, dz = 0 where r < s < R. Show that there is an analytic...
  19. F

    Calc 1 Antiderivative Practice: Solving Complex Integrals for Final Exam

    Homework Statement Hi everyone. My Calc 1 final exam is tomorrow and due to some weather related issues we were not able to cover all material for this quarter. With that said, my professor gave us a take home quiz on material that was unable to be covered. I have done by best, but I...
  20. F

    What is the general antiderivative of this complicated function?

    Homework Statement Hi everyone. My Calc 1 final exam is tomorrow and due to some weather related issues we were not able to cover all material for this quarter. With that said, my professor gave us a take home quiz on material that was unable to be covered. I have done by best, but I...
  21. M

    Antiderivative of 1.4x*cos(x^1.9)

    Question f'(x) = 1.4x*cos(x^1.9) Find f(x) Attempt Ok, first of all, I'm really bad at Calculus, so bear with me >__< I figured to find the antiderivative of the thing is equivalent to \int\left1.4x*cos\leftx^{1.9}\right\right)dx I've tried simple substitution (u = x^1.9) and that...
  22. C

    Antiderivative of square root function

    1. find the antiderivative of sqrt(81-x^2) in the bounds 0 to 9/2 okay so the first thing i did was sin substitution. where I made x = 9sin(t) dx = 9cos(t) where t is theta. then after some manipulating i got that that new integral was 9cos^2(t) i then used the double angle formula and...
  23. C

    Finding the Antiderivative of a Reciprocal: Step-by-Step Guide

    1. Find the antiderivative of 1/(x+3)^2 Okayy i knwo this is an easy problem but i COMPLETELY forget how to do it. Ive tried using partial fractions but it doesn't seem to be working. I just need to know how to start the problem then i should be alright from there. Thank you!
  24. U

    Find Anti Derivative of -5.4sin(3t) - Tips & Solutions

    Homework Statement What is the anti derivative of -5.4sin(3t)The Attempt at a Solution I have not done this in a while and i can't seem to find my prev. notes but is it not -5.4cos(3t)*3 + C
  25. G

    Proof of Antiderivative: Techniques & Math Behind x^n

    There are several techniques for derivation that have proved that the derivative for x^n is n*x^(n-1). There is the method of incrementation, working extensively with limits, etc... I'm unsure of the antiderivative. Is there any mathematical proof for the antiderivative or did they simply say...
  26. R

    Calculating the Antiderivative of (-4x)/(x^2 + 3) dx

    Homework Statement the antiderivative of (-4x)/(x^2 + 3) dx Homework Equations The Attempt at a Solution i got -2+c is that right at all, if not can somebody help me
  27. S

    What is the antiderivative of x^2 + (5/x^5) +cos9x - e^-7x?

    What is the antiderivative of x^2 + (5/x^5) +cos9x - e^-7x? I'm so lost... and also (2x + 5sqrtx)/ x^2
  28. P

    Newbie here: antiderivative for a trig function

    Greetings, Sure wish I found this forum like two years ago! Anyways, as my cherry-popping post I shall seek knowledge for the fore-going: The anti-derivative for xcos(x)? I was able to do (2x)cos(x^2) = sin(x^2), but xcos(x) has boggled me. I am not seeing it. Can anyone give me...
  29. C

    Position Function and its antiderivative

    What is the physical siginificance of a position function x(t) and its antiderivative \int x(t). Ex. a particles motion is defined by the function x(t) = x^{3} + \2x^{2} + 3x - 1 if its antiderivative is \int x(t) = \frac{x^{4}}{4} + \frac{2x^{3}}{3} + \frac{3x^{2}}{2} - x The values...
  30. F

    Heeelp with this antiderivative (silly question prob.)

    As an example for the tabular method, my book says that the antiderivative of sin 4x is -1/4 cos 4x... My calc I fundamentals are obviously rusty, so can anyone explain why it is -1/4cos 4x as opposed to just cos 4x like i thought? thank you!
  31. A

    Integral as Antiderivative: A Mathematical Proof

    Is there a mathematical proof that can prove that the integral is the antiderivative?
  32. R

    Finding antiderivative of (x^2 - 47x + 33)/10

    I need help with finding the antiderivative of 1. y=\frac{\ x^2-47x+33}{10} and 2. y=-.03(x-6)^3+x My attempts: 1. y=.1[(x^3/3)-(47x^2/2)+33x)] 2. y=(-.03x-.18)^3+x y=(\frac{\-.03x^2}{2}-.18x)^3+(x^2/2)
  33. R

    What are the antiderivatives of y=\frac{\ x^2-47x+33}{10} and y=-.03(x-6)^3+x?

    I need help with finding the antiderivative of 1. y=\frac{\ x^2-47x+33}{10} and 2. y=-.03(x-6)^3+x My attempts: 1. y=.1[(x^3/3)-(47x^2/2)+33x)] 2. y=-.03x-.18^3+x y=(\frac{\-.03x^2}{2}-.18x)^3+(x^2/2)
  34. T

    Exploring Antiderivatives: Solving for Unknown Functions

    Homework Statement Q1. http://img523.imageshack.us/img523/5933/17857219ur6.jpg Q2. http://img152.imageshack.us/img152/8133/32665282he1.jpg The Attempt at a Solution I really don't know how to go about either of the two. In the second one I know that g(0)=100, g'(0)=-10 and...
  35. A

    How do i find the antiderivative of this

    x/ square root sign(x^2 + 1) i did this x/ (x^2 + 1)^(1/2) simplified it then i did this (x^2/2)/ 3/2(x^2 + 1)^(3/2) antidirevitionated it then i cleaned it up x/ 3( x^2 + 1)^ 3 is this right
  36. O

    Antiderivative of a Quartic Function

    Homework Statement I need to find information pertaining to the antiderivative of a function. At the moment, I just need help in finding part of the derivative: what is happening at the double root (on the derivative) I am given a quartic function labelled f' with roots at: (-5,0), a double...
  37. R

    How do I find the antiderivative of(12e^2x-5)

    How do I find the antiderivative of (12e^2x-5) My attempt: u=2x-5 u'=2 12u^u x 2 24e^2x-5 What do I do next?
  38. R

    Solving for f(t) in the equation f'(t)=8cos(t)+sec^2(t), given f(pi/3)=4

    Find f if f '(t)=8cost+sec^2t and f(pi/3)=4 . so f(t)=8sint+tant+c 4=8sin(pi/3)+tan(pi/3)+c c=-3^1/2 so f(t)=8sint+tant-3^1/2 is that right
  39. M

    How Do You Solve the Integral of t * csc^2(t) dt?

    Homework Statement integral (t * csc^2 (t) ) dt Homework Equations The Attempt at a Solution t * -cot (t) - integral( 1 * -cot (t)) u= t dv= csc^2(t) du= 1 v= - cot (t) -t * cot(t) - ? I don't understand...
  40. M

    Antiderivative of Cotangent and Arcsine - Explanation Welcome

    does anyone know the antiderivative of cotangent or of arcsine? any explanation would be appreciated.
  41. S

    Finding the Antiderivative of sec (squared) x

    Homework Statement The average value of sec (squared) x over the interval 0 is less than or equal to x which is less than or equal to pie divided by 4. What is the antiderivative of sec (squared) x? Homework Equations The Attempt at a Solution average value is f(x)dx of...
  42. S

    Efficient Method for Finding the Integral of Cos^2(x)

    Homework Statement whats the integral of cos^2(x) Homework Equations ? The Attempt at a Solution is it just sin^3(x)/3?
  43. dontdisturbmycircles

    Mastering Antiderivatives: Solving Challenging Problems with U-Substitution

    Homework Statement Find \int \frac{x^{2}+3x+7}{\sqrt{x}}dx Homework Equations The Attempt at a Solution I am reviewing for a test on tuesday. I can't see a good algebraic method to solve this using the fact that \int f(g(x))g'(x)dx=F(g(x))+C so I am starting to think it...
  44. M

    How do I graph an antiderivative?

    hey all if I have the function for a derivative (which it is impossible to find an integral for, by the way), how do I graph the original? This graph is shaped kind of like an M, meaning it has some sharp points...on the original graph, would these be asymptotes or also sharp points?? thanx...
  45. B

    The antiderivative ofx*exp(1-x)

    pleasezz the antiderivative ofx*exp(1-x) hi all,,,,,, pleasezz the antiderivative ofx*exp(1-x) tomorrow 11/1/2007 is the last day please
  46. D

    How to find antiderivative of product

    problem: find the anti derivative of x^5 + tan(2x)sec(2x)dx how do you find the anti derivative of the second half of that problem tan(2x)sec(2x)
  47. S

    Antiderivative of e^-x & cos(mx): Discover the Solutions Here!

    antiderivative of e^-x I get -e^-x antiderivative of cos(mx) where m is a constant I get \frac {1}{m} \sin x Please let me know if my attempts are correct. Thanks alot!
  48. C

    One more antiderivative question.

    Hi there, I am trying to find the antiderivative of \frac{1}{2x} but can't seem to do it. I am trying to separate the terms so that I can do each antiderivative seperately, but I don't see a way to do that. My most natural first attempt is to convert it to 2x^{-1} but of course since they are...
  49. C

    How do I find the antiderivative of (x^2+1)/√x for integration?

    Hi, I am trying to integrate \int_{1}^{2} \frac{x^{2}+1}{\sqrt{x}} using the Evaluation Theorem. So my first step is to find the antiderivative of \frac{x^{2}+1}{\sqrt{x}}.. And that is where my troubles lie. I start by rewriting it as (x^{2}+1)*(x^{-1/2}} but then realize that I don't...
  50. F

    What is the Antiderivative of Secant with a Non-Integer Power?

    For one of my homework assignments, I had to find the integral of a function. I got my function simplified to sec(t)^(8/3). I tried to use the reduction formula for sec(t)^n, but I believe that it only works if the power of sec is an integer. Could somebody help me out, please? Edit: I...
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