Antiderivative Definition and 177 Threads

Problem: find $$\int \frac{dx}{1+x^{4}}.$$ Answer: It's perhaps a not-so-well-known fact that, although $x^{2}+y^{2}$ does not factor over the reals, $x^{4}+y^{4}$ does. In fact, $$x^{4}+y^{4}= \left(x^{2}+ \sqrt{2} \, xy+y^{2} \right) \left(x^{2}- \sqrt{2} \, xy+y^{2} \right).$$ Hence, we have...

1. The problem, the whole problem, and nothing but the problem $$ \int \sqrt{\frac{1+x^2}{1-x^2}} dx $$ Homework Equations ? The Attempt at a Solution I tried a partial fraction decomp, but this obviously didn't get me very far because this isn't a rational function. Is there some kind of...

1. The problem, the whole problem, and nothing but the problem \int \frac{dx}{ \sqrt{ 1+ \sqrt{ 1+ \sqrt{ x } } } } Homework Equations u-substitution (in the style of trig substitution) I think that I've got it figured out, I just don't know if my substitutions were legitimate...

Homework Statement ∫(x3 - 3x2 + x + 1)/√x dx The Attempt at a Solution ∫x3-1/2 - 3∫x2-1/2 + ∫x1-1/2 + ∫x1-1/2 ∫x5/2 - 3∫x3/2 + ∫x1/2 + ∫1/2 (x7/2)7/2 - (3x5/2)5/2 + (x3/2)3/2 + (x3/2)3/2 + C (2x7/2)/7 - (6x5/2)/5 + 2x3/2)/3 + (2x3/2)/3 + C Thank you so much!

I'm in a high school pre-calculus class and a statistics class. For the latter, we are given z-tables for some of our tests. I don't like these z-tables. Thus, I decided that a more direct approach (fundamental theorem of calculus) would be more accurate and, more importantly, more fun. My...

Homework Statement f'(u) = 1 / (1 + u^3) g(x) = f(x^2) Find g'(x) and g'(2) Homework Equations The Attempt at a Solution So the derivative of function f at u is: 1 / (1 + u^3) That means g'(x) would be f'(x^2), but to find the general derivative of f at u is 1 / (1 + u^3) so can...

G(x) is the antiderivative of F(x) with G(1) = 1; Find a formula for G(x)... Find a formula for G(x) in terms of F(x). I would have just written \int_0^x{F(x)dx} but I know the value G(1) = 1 has to be included somewhere. Can someone help me figure this one out?

The problem asks to find a formula for a higher degree antiderivative. This formula pattern is similar to the one stated in the Fundamental Theorem of Calculus: F(X)=∫f(t)dt. Fn(x)=∫*F(t)dt, with certain expression in the asterisk.

Homework Statement find the antiderivative of 1/(x^2) The Attempt at a Solution I'm pretty sure you just find the antiderivatives of the numerator and the denominator. the antiderivative of 1 is x. the antiderivative of x^2 is (x^3)/3 mutliply the numerator by the inverse of...

Homework Statement Here is the problem: ∫(1+sin(x))/(cos^2(x)) dx Also- how do you guys type equations in this? The quick symbols doesn't have a fraction bar or definite integral... Homework Equations sin^2(x)+cos^2(x)=1 The Attempt at a Solution I substituted 1-sin^2(x) for cos^2(x) and...

The derivative of x^x is x^x(lnx+1) but what would be its antiderivative? I don't think the answer is in elementary terms. According to someone there is a special function made just to answer this question. So what is it?

Homework Statement Show that the function U(x) defined by U(x)=\begin{cases} 0 & \text{ if } x<0 \\ 1 & \text{ if } x\geq0 \end{cases} has not an antiderivative in (-∞, +∞). (Suggestion: Assume U has an antiderivative F in (-∞, +∞) and obtain a contradiction, showing that this...

Homework Statement Find the antiderivative of: \int \frac{dx}{x^3 - 2x^2 + 4x - 18} Homework Equations This was asked in my Calculus II class right after we finished dealing with Solving for Integrals using Partial Fractioning. The Attempt at a Solution This is really more of an algebra...

Homework Statement v(t) = (-2 / (1 + 2t)) + 2 I must find the antiderivative s(t) such that s(0) = 5. The Attempt at a Solution u = 1 + 2t du/dt = 2 dt = du/2 √(-2 / (1 + 2t dt)) + 2 =√(-2u^-1 du/2) + 2 =√(-u^-1 du) + 2 = -ln|1+2t| + 2t + C At t = 0 -ln(1 +...

Homework Statement dy/dx = y(1-y) Homework Equations The Attempt at a Solution dx = dy(y(1-y))?

Is the antiderivative of an even function even? Specifically, given an even function g(x)=g(-x) will the following function be even? Odd? For constant a>0, r a dummy variable h(x,t) = Definite Integral [ g(r) dr, FROM x-at TO x+at] Asking whether or not h(x,t)=h(-x,t) This is the...

Homework Statement As part of a bigger problem, I am trying to find the area under the curve f(x) = sqrt{2x - x2} between x = 0 and x = 2. To do this, I have to find the antiderivative of f(x) Homework Equations antiderivative f(x) = axb F(x) = a/(b+1) * xb+1 chain rule f(x) = a(b(x)) f'(x)...

Homework Statement Find the antiderivative of \frac{1}{\sin\alpha} Homework Equations The Attempt at a Solution Ugh, maybe it's just a temporary brainfreeze, but I have no idea how to go about it.

Homework Statement a) Does f(z)=1/z have an antiderivative over C/(0,0)? b) Does f(z)=(1/z)^n have an antiderivative over C/(0,0), n integer and not equal to 1. Homework Equations The Attempt at a Solution a) No. Integrating over C= the unit circle gives us 2*pi*i. So for at least one...

Homework Statement why \frac{1}{z} don't have derivative? i know that ( \log z )'=\frac{1}{z} so why \int \frac{1}{z} \mbox{d} z don't exist? Homework Equations The Attempt at a Solution

Hi, so my question is the subject line. In the multiply connected domain |z|>0, does the function f(z) = e^z/z^3 have an antiderivative? I'm learning from Brown and Churchill, and they have a theroem on pg. 142 that leads me to believe it does. I don't remember what my prof said about this...

Homework Statement Antiderivative of sqrt[e^x+ln(x^2)+1] Homework Equations The Attempt at a Solution (2[(e^x+ln(x^2)+1)^(3/2)]x^2)/[3(e^x+2x)]

Homework Statement I'm working on an infinite series problem and need to find the antiderivative of 1/((x(lnx)^3). Homework Equations u=lnx The Attempt at a Solution I know I have to use the substitution u=lnx, but I still can't figure out what the answer is. I know the...

Hello, I'm looking for f given f'' and two conditions. [PLAIN]http://img375.imageshack.us/img375/2572/antider.jpg Going from f'' to f', I get f'(x) = 5x^4 + 4x^3 + 4x + C But with the two conditions, I feel that I cannot progress from here. I feel that one of those conditions should be...

Anybodys knows how to resolve the problem of the subject. I am working in it for three hours. I am desperated. I have tried all the subtitutions of "Calculus. Schaum. Ayres&Mendelson". Whithout suceeds. Thank you. Diego.

antiderivative of 6e^x and 4x^2 --> 6e^x -->1/6e^x?? --> 4x^2 --> 2x^3??

Homework Statement Just trying to figure out the anti-derivative of cosh(x^2). Homework Equations I knowthe antiderivative cannot be expressed as an elementary function but I am pretty clueless of getting the antiderivative though! The Attempt at a Solution I am baffled by this one...

Basically I need to know \int_0^h x\frac{1}{2\sqrt{hx}}dx my working, \frac{1}{2\sqrt{hx}} \int_0^h \frac{x}{\sqrt{x}} , how do i do this?, \int \frac{1}{\sqrt{x}} \int x^(\frac{-1}{2}) \frac{1}{\frac{-1}{2}+1} x^(\frac{-1}{2}+1) \frac{x^\frac{1/2}}{\frac{1}{2}} 2x^\frac{1}{2}...

Homework Statement \int(2x^2+1)^7 Homework Equations The Attempt at a Solution u=2x^2+1 du=4xdx u7 (1/4x)du I am stuck... I don't know what to do next...

I'm trying to find the area between the antiderivative of the curve y = 5x3 - 23x2 - x + 3 passing through the point (1, e) and the x-axis from x = 1/√2 to x = 17π/11, to 3 significant figures. I'm self teaching myself calculus, and this has me stumped.

x^2/2 is an antiderivative of x, for the derivative of x^2/2 with respect to x is x. Formally speaking, can I consider x^2/2 + y, where y is a variable and not a constant, to be an antiderivative of x, since the partial derivative of x^2/2 + y with respect to x equals x?

Hello, if I have the following unknown function f(x_1,\ldots,x_n) Assuming I am given all its partial derivatives \frac{\partial f}{\partial x_i} is it possible to get the original function f ? This is clearly possible for a one-variable function f(x). If we know df/dx we just need to...

Homework Statement http://img64.imageshack.us/img64/5430/80433637.jpg Homework Equations The Attempt at a Solution for a, would it be local.max: x=7.7 and local.min: x=4.8 ? and for b, abs.max: x=11 and abs.min: x=2 ? im not sure if i did it right but looks like this isn't...

Homework Statement $g(x)=\int _{2 }^{\sin x}\sqrt{1- t^2}dt$ whats g'(x)...Homework Equations The Attempt at a Solution how to find the antiderivative of sqrt(1-t^2)?

Homework Statement f(x)=1+e^x/1-e^xHomework Equations truth is, I don't even know how to approach this, i know i have to swap the variables but now I'm all confused because a friend told me to do this, in this particular case f(x)=[1+(e^f)-1^x]/[1-(e^f)-1^x] The Attempt at a Solution i don't...

A lot of apparently innocent elementary functions, like exp(-x^2) or (sin x)/x, have not antiderivatives in terms of elementary functions. I've read that "Differential Galois theory" explains this, and gives an algorithmic method to know if a given elementary function has or has not elementary...

I've looked through my textbook and I've found no mention, but I've seen some posts on the Internet which explicitly state to check for continuity while taking the derivative certain functions. Is it because: F'(x) = f(x) I understand that a function is required to be continuous and...

Homework Statement Decide whether f(x)=\int (1-cos(x))/x^2 is improperly integrable on (0, infinity). Homework Equations The Attempt at a Solution I understand the concept of improper integration, but I don't see how to take the antiderivative -- I tried substitution and by parts...

The proof I'm familiar with for relating the antiderivative to the area under a curve involves usage of the mean value theorem, which for that particular case, implies continuity for the curve. Thus, integration as a process for finding the area under a curve should be valid under the...

The text is giving me steps on how to do it but I don't get what's it is asking. The equation is (v^-.5)v'=k where v= volume Then says we consider the chain rule together with elementary antiderivative formulas to determine an antiderivative with respect to t of (v^-.5)v' I haven't...

Homework Statement Find the antiderivative of y=tan^2x+sec^2x Homework Equations N/A The Attempt at a Solution Seems to be a simple question, but the answer is eluding me no matter what I do. My first try was to replace the tan^2 with sec^2-1, and then factor out a 1/2 from the...

Homework Statement Evaluate the indefinite integral: \int7x+1/x^2+1 dxHomework Equations The Attempt at a Solution My first attempt at the solution was to try using substitution. I set u=x^2+1. so du=2x dx and x=sqrt(u-1). Then I rewrote the integral so it is \int7du/4usqrt(u-1). This is...

3 sin ^2 t cos t (i) + 3 sin t cos^2 t (j) + 2 sin t cost (k) I have to take the antiderivate for each Vector. Then I have to evaluate it from pi/2 to o. I'm confused because I can't use a trigonometric substitution. Cosine is odd for the I vector but I can't substitute 1-sin^2...

Homework Statement arctan(x)/(1+x2). finding the antiderivative Homework Equations arctanx=1/1+x^2 The Attempt at a Solution I tried pulling out arctan(x) S 1/(1+x^2) dx --> arctan^2(x)+c but the answer also has a 1/2 in front of the arctan...where'd that come from?

Homework Statement what is the antiderivative of e^ln(4)x Homework Equations The Attempt at a Solution Ok someone told me that the antiderivative of 4^x is e^ln(4)x. So what would be the antiderivative of e^ln(4)x? I need these antiderivatives in order to solve my homwork...

edit: nvm. i'll figure it out... :/

need help finding an antiderivative! Homework Statement 1 ∫ 36/ (2x + 1) ^3 dx 0 I just can't figure out how to take the antiderivative of this! What do I do with the (2x +1) ^3?? Homework Equations The Attempt at a Solution

Homework Statement Okay, I think I'm finally getting the hang of these antiderivatives. However, I'm still stumbling some on trigonometric functions. Find the antiderivative of f(\theta) \ = \ \frac{1 + \cos^2{\theta}}{\cos^2{\theta}} Homework Equations f(\theta) \ = \ \frac{1 +...

Homework Statement Find the antiderivative of F(x) \ = \ \sqrt{\frac{5}{x}} Homework Equations F(x) \ = \ \sqrt{\frac{5}{x}} The Attempt at a Solution 1.) F(x) \ = \ \sqrt{\frac{5}{x}} 2.) F(x) \ = \ (\frac{5}{x})^\frac{1}{2} 3.) F(x) \ = \ \frac{5^\frac{1}{2}}{x^\frac{1}{2}}...

I am suppose to evaluate this integral by using substitution f'(x)=(e^-0.5x)/(1-e^-.5x) I will be very thankful if someone were able to tell me a strategy in antiderivitating fractional problems. What I did: I set u=-0.5x dx=du/-0.5 therefore i got (-1/0.5)((e^u)/(1-e^u))du I...