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.

View More On Wikipedia.org
Recent contents Top users
  1. A

    Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x

    I proceeded as follows $$\int\frac{2(\sqrt3-1)(cosx-sinx)}{2(\sqrt3+2sin2x)}dx$$ $$\int\frac{(cos(\pi/6)-sin(\pi/6))(cosx-sinx)}{(sin(\pi/3)+sin2x)}dx$$ $$\frac{1}{2}\int\frac{cos(\pi/6-x)-sin(\pi/6+x)}{sin(\pi/6+x)cos(\pi/6-x)}dx$$ $$\frac{1}{2}\int cosec(\pi/6+x)-sec(\pi/6-x)dx$$ Which leads...
  2. KungPeng Zhou

    Evaluate Indefinite Integrals

    \frac{d^{2}}{dx^{2}}\int_{0}^{x}(\int_{1}^{sint}\sqrt{1+u^{4}}du)dt=\frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du then we let m=sinx，so x=arcsinx，then we get \frac{d}{dx}\int_{0}^{sinx}(\sqrt{1+u^{4}})du=\frac{dm}{dx}\frac{d}{dm}\int_{0}^{m}(\sqrt{1+u^{4}})du=\sqrt{1+m^{4}}\frac{dm}{dx}，then we...
  3. KungPeng Zhou

    Can't Find a Correct Method to Integrate \int (t - 2)^2\sqrt{t}\,dt?

    When I encountereD this kind of question before.For example \int x\sqrt{2+x^{2}}dx We make the Substitution t=x^{2}+2，because its differential is dt=2xdx，so we get \int x\sqrt{2+x^{2}}=1/2\int\sqrt{t}dt，then we can get the answer easily But the question，it seems that I can't use the way to...
  4. chwala

    Find the indefinite integral of the given problem

    Now the steps to solution are clear to me...My interest is on the constant that was factored out i.e ##\frac{2}{\sqrt 3}##... the steps that were followed are; They multiplied each term by ##\dfrac{2}{\sqrt 3}## to realize, ##\dfrac{2}{\sqrt 3}\int \dfrac{dx}{\left[\dfrac{2}{\sqrt...
  5. E

    MHB Indefinite integral in division form

    I have the following integration - $$\int \frac{2}{x - b \frac{x^{m - n + 1}}{(-x + 1)^m}} \, dx $$ To solve this I did the following - $$\int \frac{1 - b \frac{x^{m - n}}{(-x + 1)^m}+1 + b \frac{x^{m - n}}{(-x + 1)^m}}{x(1 - b \frac{x^{m - n}}{(-x + 1)^m})} \, dx $$ Which gives me -...
  6. brotherbobby

    The integral of a function ##f(x)## from its graph

    Problem statement : I start by putting the graph of (the integrand) ##f(x)## as was given in the problem. Given the function ##g(x) = \int f(x) dx##. Attempt : I argue for or against each statement by putting it down first in blue and my answer in red. ##g(x)## is always positive : The exact...
  7. A

    I What is the indefinite integral of Bessel function of 1 order (first k

    Hi When we find integrals of Bessel function we use recurrence relations. But this requires that we have the variable X raised to some power and multiplied with the function . But how about when we have Bessel function of first order and without multiplication? How should we integrate it ?
  8. greg_rack

    Problem solving a parametric indefinite integral

    Since ##h## and ##k## are constants: $$\frac{h}{k}\cdot \int \frac{1}{y(h-y)} \ dy$$ Now, rewriting the integrating function in terms of coefficients ##A## and ##B##: $$\frac{1}{y(h-y)}=\frac{A}{y}+\frac{B}{h-y}\rightarrow B=A=\frac{1}{h} \rightarrow$$ $$\frac{1}{h}\int \frac{1}{y}\ dy +...
  9. greg_rack

    Apparently impossible indefinite integral?

    Hi guys, I got to solve this integral in a recent test, and literally I had no idea of where to start. I thought about substituting ##tan(\frac{x}{2})=t## in order to apply trigonometry parametric equations, integrating by parts, substituting, but I always found out I was just running in a...
  10. greg_rack

    Solving an immediate indefinite integral of a composite function

    That's my attempt: $$\int (\frac{1}{cos^2x\cdot tan^3x})dx = \int (\frac{1}{cos^2x}\cdot tan^{-3}x) dx$$ Now, being ##\frac{1}{cos^2x}## the derivative of ##tanx##, the integral gets: $$-\frac{1}{2tan^2x}+c$$ But there is something wrong... what?
  11. agnimusayoti

    Indefinite integral of cross product of 2 function

    I've tried with this work in attachment. i&m not sure of my answer is correct.
  12. Adesh

    Why does dividing by ##\sin^2 x## solve the integral?

    If we look at the denominator of this integral $$\int \frac{\cos x + \sqrt 3}{1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right)} dx$$ then we can see that ## 1 + 4\sin \left(x+ \pi/3\right) + 4\sin^2 \left(x+\pi/3\right) = \left(1+2\sin\left(x+\pi/3\right)\right)^2## and ##...
  13. Michael Santos

    The indefinite integral and its "argument"

    Homework Statement The indefinite integral $$\int \, $$ and it's argument. The indefinite integral has a function of e.g ## \cos (x^2) \ ## or ## \ e^{tan (x)} \ ## If the argument of ## \cos (x^2) \ ## is ## \ x^2 \ ## then the argument of ## \ e^{tan(x)} \ ## is ## \ x \ ## or ## \ tan (x) \...
  14. donaldparida

    B Methods of integration: direct and indirect substitution

    I have seen two approaches to the method of integration by substitution (in two different books). On searching the internet i came to know that Approach I is known as the method of integration by direct substitution whereas Approach II is known as the method of integration by indirect...
  15. D

    I Is there a way to find the indefinite integral of e^(-x^2) or e^(x^2)?

    I was wandering if there is a way to understand whether it is possible to find an indefinite integral of a function. Let's say e^(-x^2) or e^(x^2)... They can't have indefinite integrals, but how can I say it? Is there a theorem or something?
  16. Q

    Mathematica Mathematica: Convolution Integral

    Hi all! I'm new to Mathematica. I have written a code for performing a convolution integral (as follows) but it seems to be giving out error messages: My code is: a[x_?NumericQ] := PDF[NormalDistribution[40, 2], x] b[k_?NumericQ, x_?NumericQ] := 0.0026*Sin[1.27*k/x]^2 c[k_?NumericQ...
  17. nightingale123

    Finding an indefinite integral

    Homework Statement Calculate the indefinite integral of the function ## \int\frac{3x^3}{\sqrt{1-x^2}}## my book gives the answer ##-(2+x^2)\sqrt{1-x^2}+C## Homework EquationsThe Attempt at a Solution So I started trying to calculate this indefinite integral by using a substitution...
  18. NickTesla

    B Indefinite Integral, doubt....

    doubt in this fraction in here Because he, simplify 12 with 4 ?? do not understand! someone could make another example ,with fraction ! Thank you!
  19. G

    Need help with this indefinite integral question please.

    Homework Statement find the following integral: cos(x/2) - sin(3x/2) dxHomework Equations I think the substitution method has to be used. Solve integrals by parts. The Attempt at a Solution Let u = x/2 cosu du/dx=1/2, I then inverted it so dx/du = 2/1 = 2 So dx=2du Now I have cosu2du Do I...
  20. Valour549

    I Is this a Definite or Indefinite Integral?

    F(x) = \int_a^x f(t) dt I have found various arguments online for both. Personally I think it's an indefinite integral because: 1) Its upper limit is a variable and not a constant, meaning the value of the integral actually varies with x. This is no different to the family of primitives...
  21. K

    Determine the indefinite integral

    Hello, Please can someone help me solve my problem. I have recently submitted my answer and had my work referred for an error. I have pictures of my question and working out, however i can not seem to post them on the page. can i email them to someone for advice/guidance Thanks
  22. Saracen Rue

    B Making a definite integral equal and indefinite integral?

    I have a calculator which allows me to sketch indefinite integrals - it assumes c = 0. However, when I try to use Desmos Online Graphing Calculator, it won't let me do this with it's integral function. It keeps trying to make me use definite integrals. I know that ∫(a,b,f(x)dx = F(a) - F(b), so...
  23. P

    MHB Kishan's question via email about an indefinite integral

    What is the $\displaystyle \begin{align*} \int{ \frac{54\,t - 12}{\left( t- 9 \right) \left( t^2 - 2 \right) } \,\mathrm{d}t } \end{align*}$ We should use Partial Fractions to simplify the integrand. The denominator can be factorised further as $\displaystyle \begin{align*} \int{ \frac{54\,t -...
  24. P

    MHB Effie's question via email about an indefinite integral.

    What is the indefinite integral (with respect to t) of $\displaystyle \begin{align*} 50\,t\cos{ \left( 5\,t^2 \right) } \end{align*}$? $\displaystyle \begin{align*} \int{ 50\,t\cos{\left( 5\,t^2 \right) } \,\mathrm{d}t } &= 5\int{ 10\,t\cos{ \left( 5\,t^2 \right) }\,\mathrm{d}t } \end{align*}$...
  25. G

    Finding the indefinite integral of sin^2(pi*x) cos^5(pi*x)

    Homework Statement ∫(sin2(πx)*cos5(πx))dx. Homework Equations Just the above. The Attempt at a Solution I have no idea how pi effects the answer, so I basically solved ∫(sin2(x)^cos5(x))dx. ∫(sin2(x)*cos4(x)*cos(x))dx ∫sin2(x)*(1-sin2(x))2*cos(x))dx U-substitution u = sin x du =...
  26. I

    Calculate indefinite integral using Fourier transform

    Homework Statement Use the Fourier transform to compute \int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx Homework Equations The Plancherel Theorem ##||f||^2=\frac{1}{2\pi}||\hat f ||^2## for all ##f \in L^2##. We also have a table with the Fourier transform of some function, the ones of...
  27. SteliosVas

    Indefinite integral and proving convergence

    Homework Statement okay so the equation goes: ∫(x*sin2(x))/(x3-1) over the terminals: b= ∞ and a = 2 Homework Equations Various rules applying to the convergence or divergence of integrals such as the p-test, ratio test, squeeze test etc The Attempt at a Solution Okay so I have tried...
  28. Chip

    Solving the Indefinite Integral for <xp>=<px>*

    First of all, I'm new here, so please bear with me if the answer to my question can be found elsewhere, but I have been working a problem and searching for an answer for two weeks now without a complete solution. In Eisberg and Resnick chapter 5, problem 15, an essential part of the problem is...
  29. Drakkith

    Evaluating an Indefinite Integral using Substitution

    Homework Statement Evaluate the Integral: ∫sin2x dx/(1+cos2x) Homework EquationsThe Attempt at a Solution I first broke the numerator up: ∫2sinxcosx dx /(1+cos2x) 2∫sinxcosx dx /(1+cos2x) Then I let u = cosx so that du = -sinx dx -2∫u du/(1+u2) And now I'm stuck. I thought about turning...
  30. S

    Finding the Indefinite Integral: Can Multiplying a Constant Change the Solution?

    I have posted my attempt and the problem above. Please help! Thanks in advance!
  31. P

    Indefinite integral with discontinuous integrand

    Suppose ##f## is defined as follows: ##f(x) = 1## for all ##x ≠ 1##, ##f(1) = 10##. Is the indefinite integral (or the most general antiderivative) of ##f## defined at ##x = 1##? I'm asking this question because I already know how to deal with, say, ##\int_0^2 f##; ##f## has only one removable...
  32. Physics-UG

    Indefinite Integral with integration by parts

    Homework Statement Evaluate ∫e-θcos2θ dθ Homework Equations Integration by parts formula ∫udv = uv -∫vdu The Attempt at a Solution So in calc II we just started integration by parts and I'm doing one of the assignment problems. I know I need to do the integration by parts twice, but I've hit...
  33. PWiz

    Indefinite integral of arcsec(x)

    Just for fun, I tried this rather trivial problem, but I think I went wrong somewhere: $$\int arcsec(x) \ dx$$ Let ##arcsec(x)=y## . Then ##x=sec \ y##, or ##y=arccos(\frac 1{x})## So the problem becomes $$\int arccos(\frac 1 {x}) \ dx$$ Let ##\frac 1 {x} = cos \ u## , so that ##dx = secu \ tanu...
  34. B

    Indefinite Integral: How to Use Trig Substitution?

    Homework Statement Find the indefnite integral using trig substitution. ∫[(x^2) / (1+x^2)]dx Homework Equations --- The Attempt at a Solution
  35. S

    How i solve this indefinite integral?

    1. http://www.imageurlhost.com/images/cnj1t05jh6e4fxqy4i5_integral.png I know that this integral is solved by the sustitution method The Attempt at a Solution I tried converting the integral to the form of Arctanx, but that x2 on the numerator ruined everything. Thanks
  36. karush

    MHB Indefinite integral using trig substitutions

    $\int\frac{1}{\sqrt{2+3y^2}}dy$ $u=\sqrt{3/2}\tan\left({\theta}\right)$ I continued but it went south..
  37. TheDemx27

    Indefinite Integral in Programming

    I've been contributing to an open source calculator, and I wanted a way to take integrals of functions. I suppose you could implement a definite integral function by using Riemann Sums, but I can't find any way to implement indefinite integrals (or derivatives for that matter). I've heard that...
  38. PWiz

    Solving Integral Problem: Arc Length of Sine

    Homework Statement I was trying to find a general expression for the arc length of sine, i.e. ##\int \sqrt{1+cos^2 x} dx##, but got stuck. I made this problem up, and with some google searches realized that it uses something known as an elliptical integral. How do I go about using it here? 2...
  39. J

    MHB How can we evaluate this indefinite integral of a definite integral?

    Evaluation of Indefinite Integral $\displaystyle \int_{0}^{1} \sqrt{1-2\sqrt{x-x^2}}dx$ $\bf{My\; Try::}$ We can write the given Integral as $\displaystyle \int_{0}^{1}\sqrt{\left(\sqrt{x}\right)^2+\left(\sqrt{1-x}\right)^2-2\sqrt{x}\cdot \sqrt{1-x}}dx$ So Integral Convert into...
  40. karush

    MHB Indefinite integral complete square

    $$\int_{}^{} \frac{1}{\sqrt{16+4x-2x^2}}\,dx$$ $$\frac{\sqrt{2}} {2}\int_{}^{} \cos\left(\frac{x-1}{3}\right)\,dx$$ So far ? Not sure
  41. karush

    MHB Evaluating Integral $$\int \frac{e^{2x}}{u} du$$

    $$\int_{}^{} \frac{e^{2x}}{e^{2x}-2}dx. \\u=e^{2x}-2\\du=e^{2x}$$ Now what?
  42. J

    MHB Integral Evluation: $$\displaystyle \int\frac{5x^3+3x-1}{(x^3+3x+1)^3}dx$$

    Evaluation of $$\displaystyle \int\frac{5x^3+3x-1}{(x^3+3x+1)^3}dx$$$\bf{My\;Try::}$ Let $\displaystyle f(x) = \frac{ax+b}{(x^3+3x+1)^2}.$ Now Diff. both side w. r to $x\;,$ We Get$\displaystyle \Rightarrow f'(x) = \left\{\frac{(x^3+3x+1)^2\cdot a-2\cdot (x^3+3x+1)\cdot (3x^2+3)\cdot...
  43. G

    Can You Solve This Strange Indefinite Integral?

    Hello, everyone I've trying to solve this integral but it seems like the methods I know are not enogh to solve it. So I'd be glad if you could give me some trick to get into the answer. Here it is: Thanks in advance!
  44. P

    The Indefinite Integral of Zero

    Argument A ##∫ 0 dx = 0x + C = C## Argument B ##∫ 0 dx = ∫ (0)(1) dx = 0 ∫ 1 dx = 0(x+C) = 0## Discuss.
  45. J

    MHB Indefinite Integral question

    \displaystyle \int e^{x^4}\left(x+x^3+2x^5\right)\cdot e^{x^2}dx =
  46. P

    MHB Johnsy's question about finding an indefinite integral

    The big clue here is the square root in the denominator, because $\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left( \sqrt{x} \right) = \frac{1}{2\,\sqrt{x}} \end{align*}$. So this suggests that you probably need to find a square root function to substitute. Rewrite your...
  47. J

    MHB Evaluation of Indefinite Integral

    Calculation of \(\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx\) **My Try \(\displaystyle :: I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin x\cdot \sqrt{\cos 2x}}\cdot \frac{\sin x}{\sin x}dx = \int\frac{\sqrt{2\cos^2 x-1}}{\left(1-\cos^2 x\right)\cdot \sqrt{2\cos^2...
  48. J

    What is the indefinite integral of cosecant function?

    What is the indefinite integral of cosec(\theta)?
  49. J

    Indefinite Integrals of Scalar and Vector Fields: A Path Independence Dilemma?

    Is possible to compute indefinite integrals of functions wrt its variables, but is possible to compute indefinite integrals of scalar fields and vector fields wrt line, area, surface and volume?
  50. L

    MHB Help with Solving Indefinite Integral

    Hi, I tried to solve this integral \int\sqrt{1-\frac{1}{x^3}}dx but i can't solve it... can someone help me?
Back
Top