Indefinite integral Definition and 207 Threads

Is this possible.. Say, a∫b f(x)dx = G(x)|x=b - G(x)|x=a = S, where S, a and b are known. Can we find G(x) ?

Evaluation of $\displaystyle \int\sqrt\frac{1+\tan x}{\csc^2 x+\sqrt{\sec x}}dx$ I have Tried The Given Integral Using $\displaystyle \tan x = \frac{2\tan \frac{x}{2}}{1-\tan^2 \frac{x}{2}}$ and $\displaystyle \cos x = \frac{1-\tan^2 \frac{x}{2}}{1-\tan^2 \frac{x}{2}}$ and $\displaystyle \sin x...

$\displaystyle \int \frac{\ln\left(x^2+2\right)}{(x+2)^2}dx$ $\bf{My\; Try::}$ Given $\displaystyle \int \ln \left(x^2+2\right)\cdot \frac{1}{(x+2)^2}dx$ Using Integration by parts, we get $\displaystyle = -\ln\left(x^2+2\right)\cdot \frac{1}{(x+2)} + 2\int \frac{x}{\left(x^2+2\right)\cdot...

Homework Statement Evaluate the indefinite integral of x*cos(3x)^2 Homework Equations Integration by parts: \int(udv)= uv - \int(vdu) The Attempt at a Solution Im having trouble finding the antiderivative of cos(3x)^2 (which I designated as dv when doing integration by parts). I...

How does \int \frac{2x + 2}{x^2 + 2x + 5} \, dx turn into \ln(x^2 + 2x + 5)? How are they getting rid of the numerator are they just dividing by the reciprocal of 2x + 2?

Okay so I'm working on this problem \int \frac{x^2}{\sqrt{4 - x^2}} \, dx I do a substitution and set x={\sqrt{4}}sinu I get to this step fine \int 4sin(u)^2 I know that u = arcsin(x/2) so I don't see why I can't just substitute in u into sin(u)? I tried this and I got \int 4 *...

I understand why a definite integral of the form ^{b}_{a}∫ƒ(x)dx has the differential dx in it. What I don't understand, and what my teacher hasn't explained is why an indefinite integral (i.e. an antiderivative) requires the differential. Why does ∫ƒ(x)dx require that dx to mean...

Here is the question: I have posted a link there to this thread so the OP can view my work.

Here is the question: I have posted a link there to this thread so the OP can view my work.

I this old thread it mentions that the indefinite integral of f'(x)/f(x) is log(|f(x)|)+C which means that there is some ambiguity about the sign of f(x). There does however, seem to be no ambiguity about the value of C as it always appears to be zero, but I have never seen this mentioned...

I am trying to find the following indefinite integral: Homework Statement ∫\sqrt{}x/(x-1)dx Homework Equations None The Attempt at a Solution I tried to use substitution but got nowhere. I set u=\sqrt{}x so du=1/(2\sqrt{}x)dx. However from here on on I got stuck. I also tried...

Here is the question: I have posted a link there to this thread so the OP can see my work.

Here is the question: I have posted a link there to this topic so the OP can see my work.

$\displaystyle \int\frac{1}{(a+b\sin x)^2}dx$, where $a>b$ My Trial :: Using Integration by parts:: $\displaystyle \int\frac{1}{(a+b\sin x)^2}dx = \frac{1}{b}\int -\csc (x)\cdot \frac{-b\sin x}{(a+b\sin x)^2}dx$ $\displaystyle -\frac{1}{b}\cdot \csc (x)\cdot \frac{-1}{(a+b\sin x)}+\int...

I'm trying to integrate \int e^{4\ln{x}}x^2 dx I can't see using u-substition, x^2 isn't the derivative of e^{4\ln{x}} nor vice-versa. I tried integrating by parts and that didn't work. I used u=e^{4\ln{x}} and dv=x^2 dx I know I can't rewrite e^{4\ln{x}} as e^4e^\ln{x}

how would start solving this $\displaystyle\int\frac{dy}{1+e^y}$

Problem: Rewrite the indefinite integral ## \iint\limits_R\, (x+y) dx \ dy ## in terms of elliptic coordinates ##(u,v)##, where ## x=acosh(u)cos(v) ## and ## y=asinh(u)sin(v) ##. Attempt at a Solution: So would it be something like, ## \iint\limits_R\, (x+y) dx \ dy =...

Homework Statement Sorry for the poor use of Latex, I have tried to get it to work but it seems to never come out as I would like. Using a trigonometric or hyperbolic substitution, evaluate the following indfenite integral, ∫\frac{1}{\sqrt{(x^2-1)^5}} dx Homework Equations I...

how would i go about solving this $\displaystyle\int\frac{\arctan x}{1+x^2}dx$? i tried substitution but i didn't work.

Here is the question: I have posted a link there to this topic so the OP can see my work.

Here is the question: I have posted a link there to this topic so the OP can see my work.

Here is the question: I have posted a link there to this topic so the OP can see my work.

This is the integral I am trying to evaluate. I would very much appreciate any help. \[\int \frac{2x^3-1}{x+x^4}dx\] MY APPROACH: \[\int \frac{2x^3-1}{x+x^4}dx\] \[\int \frac{1}{2}.(\frac{4x^3+1}{x^4+x}-\frac{3}{x^4+x})dx\] \[\frac{1}{2}\log{x+x^4}-\frac{1}{2}\int \frac{3}{x^4+x})dx\] now we...

Homework Statement Ok the problem is: ∫-1/(4x-x^2) dx The answer in the back of the book is: (1/4)ln(abs((x-4)/x))) + C Homework Equations I think this would be used somehow: ∫ du/(a^2-u^2) = 1/2a ln(abs((a+u)/(a-u))) + C The Attempt at a Solution ∫-1/(4x-x^2) dx...

Here is the question: I have posted a link there to this topic so the OP can see my work.

Homework Statement \int \left(\frac{x}{x\cos x-\sin x} \right)^2 dxHomework Equations The Attempt at a Solution Factoring out ##\cos x## from the denominator, the integral transforms to \int \sec^2x \left(\frac{x}{x-\tan x}\right)^2dx Substituting ##\tan x=t##, ##\sec^2 xdx=dt## \int...

Homework Statement \int \sqrt{\frac{\csc x-\cot x}{\csc x+\cot x}} \frac{\sec x}{\sqrt{1+2\sec x}}dxHomework Equations The Attempt at a Solution The integral can be simplified to: \int \sqrt{\frac{1-\cos x}{1+\cos x}} \frac{1}{\sqrt{\cos x} \sqrt{\cos x+2}}dx Using ##\cos...

Homework Statement \int1/(1+\sqrt{2x})\,dx Homework Equations u=1+\sqrt{2x} \Rightarrow \sqrt{2x}=u-1 du=1/\sqrt{2x}dx \Rightarrow \sqrt{2x}du=dxThe Attempt at a Solution \int1/(1+\sqrt{2x})\,dx = \int\sqrt{2x}/(1+\sqrt{2x})\,du = \int(u-1)/u\,du = \int\,du-\int1/u\,du = u-ln|u|+C =...

Here is the question: Here is a link to the question: Find the indefinite integral? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Evaluate the indefinite integral as an infinite series ∫ sin(x2) dx Homework Equations The Macluarin series of sin x = ∞ Ʃ (-1)nx2n+1/(2n+1)! n=0 The Macluarin series for sin(x2) = ∞ Ʃ (-1)x4n+2/(2n+1)! n=0 The Attempt...

Here is the question: Here is a link to the question: Integral of (e^sqrt(x) + 1) with respect to x? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Here is the question: Here is a link to the question: Calc 2 integral question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Hello everyone.. Homework Statement ∫√((1+(e^-x))^2)dx 2. The attempt at a solution I first tried to do a u sub and then attempt a trig sub however I can't do anything with the e^-x left in the u sub. Does anyone have another way I can integrate this thing?? Thank you for any suggestions/help!

Here is the question: Here is a link to the question: Integration question!? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Integrate (3x^2-10)/(x^2-4x+4) dx using partial fractions. Homework Equations None The Attempt at a Solution I tried using A/(x-2) + B/(x-2)^2 but I didnt get a coeffecient of an x^2. I've also tried using (Ax+B)/(x-2) + C/(x-2)^2 Though I...

Homework Statement ∫▒〖(4x^2+2√x+1)/(2x√x) dx〗 Homework Equations The Attempt at a Solution ∫▒〖(4x^2+2√x+1)/(2x√x) dx〗 ∫▒((4x^2)/(2x^(3/2) )+(2√x)/(2x^(3/2) )+1/(2x^(3/2) ))dx 2∫▒x^(1/2) dx+∫▒〖x^(-1) dx〗+1/2 ∫▒x^(-3/2) dx 2×2/3 x^(3/2)+ln⁡x+1/2×(-2) x^(-1/2)+C 4/3...

Homework Statement I do not know how to solve the following indefinite integral. I personally think it is very difficult and would appreciate it had someone can explain it step by step? Homework Equations / The Attempt at a Solution This integral must been solved by mix of...

Homework Statement Find the indefinite integral: ∫ Cos32x Sin22x dx Homework Equations None required The Attempt at a Solution Lost on where to start. If someone could just start me off.

Homework Statement F(x)= (3^x)(e^x)dx Homework Equations F(u)=U^n=(U^(n+1))/n+1 The Attempt at a Solution I said it equaled: ((3^(x+1))/(x+1))(e^x)

$\displaystyle \int\sqrt{\frac{x^2+1}{x^2-1}}dx$

Hi, I've been wondering this since I started learning integration. I get that ∫ is basically an elongated S for "sum", because that is what it is basically doing. But then Ʃ does the same thing as well. If I'm understanding the difference, it is that Ʃ increments by finite measures, whereas ∫...

$\displaystyle \int\frac{1}{\sqrt[4]{1-x^4}}dx$

Homework Statement Evaluate the integral: ∫ {√[(a^2)-(x^2)] / (b-x)} dx Homework Equations ∫ u dv = uv - ∫v du The Attempt at a Solution I've tried using integration by parts but it makes the integral even more complex. I also tried using the table of integrals to find a solution to no...

Homework Statement Compute the following: \int \frac{cosh(x)}{cosh^2(x) - 1}\,dx Homework Equations \int cosh(x)\,dx = sinh(x) + C The Attempt at a Solution I had no clue where to start, so I went to WolfRamAlpha, and it used substitution but it made u = tanh(\frac{x}{2})...

Homework Statement Find the function f(x) such that f''(x) = \frac{1}{x^2}, f(1) = 0 and f(e) = 0 Homework Equations \int f''(x)\,dx = f'(x) + c \int f'(x)\,dx = f(x) + cx + C The Attempt at a Solution f''(x)= \frac{1}{x^2} f'(x)= \int \frac{1}{x^2}\,dx = \frac{-1}{x} + c f(x) = -\int...

EDIT: Problem found. This thread can now be ignored.Homework Statement Find the indefinite integral. Homework Equations ((y^2-1)/y)^2 dy The Attempt at a Solution I've attempted a few things. I first attempted to split the statement inside the outer parentheses into two fractions; (y^2/y...

Hello, I have recently encountered an integral that I have been able to evaluate in a sick, unholy way, and for making a proof much more elegant I would like a simple way to evaluate the integral from 0 to infinity of ln(x)/(e^x+1) . thank you!

Homework Statement Calculate the following integral: \int{\frac{\sqrt{x+1}}{x+5}dx} \ , x ≥ 1 By using the following substitution: t=\sqrt{x+1} Homework Equations Well using the integration by substitution formula. The Attempt at a Solution So I have t=\sqrt{x+1}...

Homework Statement Given:lim_{n\rightarrow ∞} \int^{a^n}_{1} \frac{t^{1/n}}{(1+t)t} dt=\int^{∞}_{1} \frac{1}{(1+t)t} dt a - Natural number. I need to prove that I can bring limit under the integral sign. Homework Equations The Attempt at a Solution I've got this so far: | \int^{a^n}_{1}...

Homework Statement ∫8x3e-cos(x4+4)sin(x4+4)dx Homework Equations Let u = cos(x4+4) The Attempt at a Solution I know the answer does not have the sin in it and only the e remains, because when the integral is found e stays unchanged. I could find somewhere online to calculate it...