 # Boost Your Skills in Finding Antiderivatives

Recent questions in Antiderivatives Cierra Mclaughlin 2023-02-12

## The antiderivative of cos x is. A)sin x B)cos x C)-sin x D) tan x Omari Mcclure 2023-02-01

## What is the antiderivative of cos x?A) sin xB) cos xC)- sin xD) tan x Taniyah Hartman 2023-01-14

## Let F(x) be an antiderivative of $\frac{2{\left(\mathrm{ln}x\right)}^{4}}{3x}$, If F(2)=0, then F(8)=? Taylor Barron 2022-11-18

## Antiderivative of a function arised in KdV equationI am computing the third non-trivial conservation law of KdV equation ${u}_{x}+6u{u}_{x}+{u}_{xxx}=0$ via the power series expansion method (Here we consider real-valued solutions only).I was given an equivalent form of the PDE:${\left(2{u}^{3}+5{u}_{x}^{2}\right)}_{t}+\left(36{u}^{3}{u}_{x}+6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}\right)$To finish the job one needs to express$\left(36{u}^{3}{u}_{x}+6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}\right)$ in a form $\left(\cdots {\right)}_{x}$It is clear that $\left(9{u}^{4}{\right)}_{x}$ is an antiderivative of $36{u}^{3}{u}_{x}$, but what is the antiderivative of$6{u}^{2}{u}_{xxx}+10{u}_{x}{u}_{xxxx}+60{u}_{x}^{3}+60u{u}_{x}{u}_{xx}$ in terms of derivatives of u? Barrett Osborn 2022-11-03

## Clarification about the Antiderivative of ${x}^{-1/3}$.In some textbook the antiderivative of ${x}^{-1/3}$ is written as$\int {x}^{-1/3}\mathrm{d}x=\frac{3}{2}{x}^{2/3}+C,$where C is a constant. But should not the following function also be considered as an antiderivative of ${x}^{-1/3}$?When ${C}_{0}\ne {C}_{1}$, F(x) cannot be written as $\frac{3}{2}{x}^{2/3}+C$.Edit: I would like to clarify: which one of the following should be the correct answer to $\int {x}^{-1/3}\mathrm{d}x$:- $\frac{3}{2}{x}^{2/3}+C$- or Kendrick Finley 2022-10-18

## Antiderivative of: $t↦\left(1-{t}^{2}{\right)}^{\lambda }$I would like to find an antiderivative of the function $t↦\left(1-{t}^{2}{\right)}^{\lambda }$ where $\lambda \in {\mathbb{R}}_{>0}$.I really don't know how to proceed. One idea is to use the generalized binomial theorem to get:$\left(1-{t}^{2}{\right)}^{\lambda }=\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{\lambda }{k}\right)\left(-1{\right)}^{k}{t}^{2k}$And by termwise integration I get that a possible antiderivative is $\sum _{k=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0}{}{\lambda }{k}\right)\left(-1{\right)}^{k}\frac{{t}^{2k+1}}{2k+1}$.The problem is that this form isn't really helpful. So is there a close form of this? So that I can study the behavior of the function when $\lambda \to \mathrm{\infty }$ on [0,1], for example. podvelkaj8 2022-10-18

## Equality of two trigonometric integrals on [0,1]I need to show, that:But frankly I cannot see way to solve it. The right-side integral is improper and as far I know both of them don't have the elementary antiderivatives. joyoshibb 2022-08-10

## Verifying Fourier series of antiderivative of a functionThe function is $f\left(x\right)=\left\{\begin{array}{ll}0,& -\pi \le x\le 0\\ x,& 0.The Fourier series of the function is:$\frac{1}{2}\left(\frac{\pi }{2}\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}\mathrm{cos}\left(nx\right)+\frac{\left(-1{\right)}^{n+1}}{n}\mathrm{sin}\left(nx\right)$.Since the function can be integrated termwise, the Fourier series for the antiderivative ${\int }_{-\pi }^{x}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}dx$ is:$\frac{1}{2}\left(\frac{\pi }{2}\right)\left(x+\pi \right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\left(\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}\mathrm{sin}\left(nx\right)+\frac{\left(-1{\right)}^{n}}{n}\left(\mathrm{cos}\left(nx\right)-\left(-1{\right)}^{n}\right)\right)$The antiderivative of the function is $F\left(x\right)=\left\{\begin{array}{ll}0,& -\pi \le x\le 0\\ \frac{{x}^{2}}{2},& 0. The Fourier series of the antiderivative is:$\frac{1}{2}\left(\frac{{\pi }^{2}}{6}\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\left(\left(\frac{\left(-1{\right)}^{n}-1}{\pi {n}^{2}}-\left(-1{\right)}^{n}\frac{\pi }{2}\right)\mathrm{sin}\left(nx\right)+\frac{\left(-1{\right)}^{n}}{n}\mathrm{cos}\left(nx\right)\right)$Where I am missing? Why I am having wrong answer. Even if I did some mistakes in calculations, what's with x term obtained in piece wise integration. Thanks in advance. Max Macias 2022-08-06

## Finding the value of b so ${\int }_{1}^{b}\left(x-2{\right)}^{3}dx=0$.Please how do find $b>1$ so that This question is on a chapter dealing with antiderivatives and I'm not sure how to go about it. At this point it is assumed that I don't know how to integrate yet. I'm also not allowed to use the fundamental theorem of calculus. wendi1019gt 2022-08-04

## A ball with mass 0.15kg is thrown upwardswith initial velocity 20 m/sec from the roof of a building 30mhigh. There is a force due to air resistance of ${v}^{2}/1325$.where the velocity is measured in m/sec.(a) Find max height above ground ball reaches.(b) Find the time that the ball hits the ground. comAttitRize8 2022-07-20

## Find the antiderivative of $\sqrt{3x-1}dx$.I got $\frac{2}{3}\left(3x-1{\right)}^{3/2}+c$ but my book is saying $\frac{2}{9}\left(3x-1{\right)}^{3/2}+c$.Can some one please tell me where the 2/9 comes from? Alexandra Richardson 2022-07-17

## How to prove the function f has an antiderivative?There is a question in our analysis book and I have lots of problem with it. It says that:set $f\left(x\right)=\left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ \mathrm{sin}\left(\frac{\pi }{x}\right)\phantom{\rule{1em}{0ex}}& x>0\end{array}$$g\left(x\right)=\left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ 1\phantom{\rule{1em}{0ex}}& x>0.\end{array}$prove that f has an antiderivative but g does not.My first problem is about antiderivative of g. I think it has an antiderivative and it is$\begin{array}{rl}G\left(x\right)=& \left\{\begin{array}{ll}0\phantom{\rule{1em}{0ex}}& x\le 0\\ x\phantom{\rule{1em}{0ex}}& x>0.\end{array}\end{array}$Why we can't say that G is antiderivative of g? My second problem is about finding the antiderivative of f. As you may know antiderivative of $\mathrm{sin}\left(\frac{\pi }{x}\right)$ can not be shown by the elementary functions. So, to prove that f has antiderivative I can't find a function like F which its derivative is f and I need to use another approach to prove it, but I don't have any idea about what should I do? slijmigrd 2022-07-15

## Order of antiderivatives of Schwartz functionsFor a Schwartz function $f\in \mathcal{S}\left(\mathbb{R}\right)$ it is known that$\mathrm{\exists }F\in \mathcal{S}\left(\mathbb{R}\right):{F}^{\prime }=f\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f\left(x\right)dx=0$in this case $F\left(x\right)={\int }_{x}^{+\mathrm{\infty }}f\left(t\right)dt$, see related question here.Question: Does there exist $f\in \mathcal{S}\left(\mathbb{R}\right)$ with ${f}^{\left(-k\right)}\in \mathcal{S}\left(\mathbb{R}\right)$ for all $k\ge 0$ (my guess would be no)I.e. for a given f we can test if it has an antiderivate F in the Schwartz space by simply checking if its mean is 0. Then we can do the same with F. Does this process always stop at some point?My work: It seems that all functions of the type $f\left(x\right)=p\left(x\right){e}^{-\alpha {x}^{2}}$ with p polynomial only have antiderivatives of finite order in $\mathcal{S}\left(\mathbb{R}\right)$. (I think I can prove this). As one can construct an orthonormal basis of $\mathcal{S}\left(\mathbb{R}\right)$ with such functions, e.g. the Hermite functions, I would guess that there is no function with antiderivates of arbitrary order in $\mathcal{S}\left(\mathbb{R}\right)$.Remark: If there is such an f, then $\left({x}^{k}\star f\right)=0$ for all $k\ge 0$; then via Fourier transform, this would imply that in the sense of distributions:$0=\mathcal{F}\left[{x}^{k}\star f\right]=\mathcal{F}\left[{x}^{k}\right]\mathcal{F}\left[f\right]=\left(\frac{i}{2\pi }{\right)}^{k}{\delta }^{\left(k\right)}\left(w\right)\stackrel{^}{f}\left(w\right)$ auto23652im 2022-07-14

## The extension of this derivative notation ${f}^{\left(0\right)}=f,{f}^{\left(1\right)}={f}^{\prime },...$Just a quick question on this notation, is ${f}^{\left(-1\right)}$ used for antiderivatives? Lena Bell 2022-07-13

## Is there any elementary function whose antiderivative contains an exact constant?Let's say we have . Now it's obvious to me why F is the class of functions whose derivative yields f(x). However, I was curious if it is possible for the antiderivative to be something such as $F\left(x\right)=\mathrm{tan}\left(x\right)+5+C$ as an example. I can see how we would merge this to become just $F\left(x\right)=\mathrm{tan}\left(x\right)+C$ but I'm wondering if it is possible for an elementary antiderivative to contain a constant that the entire class of antiderivatives share. My feeling is that this would never happen but I can't seem to figure out exactly why. Jamison Rios 2022-07-10

## Continuous complex function without antiderivativeIt's a well-known result that every real continuos function has an antiderivative. Is this theorem still true for a complex function? If not, can someone point out a counter-example (and proof that it is indeed a counter-example)? dream13rxs 2022-07-10

## Antiderivatives - part c, solving for x$MR\left(x\right)=4x\left({x}^{2}+26,000{\right)}^{-2/3}$I'm already lost at the part $2\int {u}^{-2/3}$. How did they get $6{u}^{1/3}+C$.a)Find the revenue functionb) What is the revenue from selling 250 ​gadgets?​c) How many gadgets must be sold for a revenue of at least ​\$50​,000?Solve for x. (How?)$6\left({x}^{2}+26,000{\right)}^{1/3}-150=50$ Dayanara Terry 2022-07-09

## Antiderivative for $\mathrm{sin}\left({t}^{2}\right)/2$?I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if someone could explain how to solve this problem.The acceleration of a particle moving along the line is given by $a\left(4\right)=t\mathrm{cos}\left({t}^{2}\right)$. If at time $t=0$, its velocity is 2m and position is 4m, what is the position of the particle at time $t=7$?(Calculator question; the correct answer is 18.303m .I got that the equation for the velocity ($v=\mathrm{sin}\left({t}^{2}\right)/2+2m$) but I can't seem to find its antiderivative and get the equation for the position. gaiaecologicaq2 2022-07-09
## Find The Antiderivative of more complex expressions.Need help finding the antiderivative of this one in particular $\frac{dx}{3x²+1}$.Would you kindly also give some tips on how to find the antiderivative of any expressions. I am having problems switching from differential calculus to integral calculus as finding the antiderivative is not as straight forward as finding the differential of an expression. mistergoneo7 2022-07-09