3
votes
0answers
52 views

Evaluate $\displaystyle\int{\frac{x^3-1}{x\sqrt{x^4+1}}}\, dx$ [on hold]

Evaluate the following indefinite integral. $$ I=\int{\frac{x^3-1}{x\sqrt{x^4+1}}}\, dx. $$
0
votes
1answer
41 views

Don´t know how to start proving this formula.

\begin{equation*} \int \frac{\cos ^{m}x}{\sin ^{n}x}dx=-\frac{\cos ^{m+1}x}{(n-1)\sin ^{n-1}x}- \frac{m-n+2}{n-1}\int \frac{\cos ^{m}x}{\sin ^{n-2}x}dx+C,\qquad (n\neq 1). \end{equation*} I`d like to ...
0
votes
1answer
48 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
1
vote
0answers
29 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
3
votes
2answers
79 views

Evaluate $\int x \sqrt{1 - x^4} \,\mathrm{d}x$

I have the following question $$\int x \sqrt{1 - x^4} \,\mathrm{d}x$$ I know we have to use trig. substitution for this and therefore, I did the following by letting $x = \sin \theta$ and $dx = \cos ...
3
votes
2answers
52 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
4
votes
0answers
63 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
0
votes
1answer
66 views

Integral of $\sin|x|$

$$\int\sin|x|~dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x)~dx+\int_0^\infty\sin x~dx$$ Left side of this sum is equals to right side, so we ...
1
vote
4answers
73 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
3
votes
1answer
64 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
0
votes
2answers
78 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [closed]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
2
votes
2answers
55 views

Reduction formula for $\int \frac{dx}{x^n \sqrt{ax+b}}$

I want a reduction formula for $$I_n=\int\frac{dx}{x^n \sqrt{ax+b}}$$ in terms of $I_{n-1}$. I have tried various substitutions but I just can't seem to find the right one. Any help or hints will ...
2
votes
2answers
158 views

What is the easiest way to integrate $y=\frac {x+4}{\sqrt{-x^2-2x+3}}$?

What is the easiest way to integrate $y=\frac{x+4}{\sqrt{-x^2-2x+3}}$ ? I tried to integrate it by making numerator in form: $-2x-2$ and then pulling it under differential, but the result drastically ...
7
votes
2answers
167 views

Fun Integral $ \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}$

$$ I\equiv \int \frac{dx}{\cos^3 x+2\sin(2x)-5\cos x}. $$ This integral does have a closed form. I am not sure where to start. We can factorize the denominator as $$ \cos^3 x+2\sin(2x)-5\cos ...
0
votes
2answers
80 views

Integral $\int\sqrt{\sin2x}\operatorname d\!x$

I tried all substitutions but failed. I need assistance to evaluate that indefinite integral. $\int\sqrt{\sin2x}\operatorname d\!x$
0
votes
1answer
26 views

Сhange the order of integration in the double integral

I have to change the order of integration in this double integral I've decided to divide it in two similar areas D1 and D2 And I've got the following result Can You chech it and state my ...
1
vote
4answers
199 views

What is the value of $\int x^x~dx$?

I am struggling with this puzzle. Question 1. Is it possible to determine the value of the indefinite integral $\int x^x~dx$ explicitly? By "explicit" I mean without power series. Question 2. What ...
3
votes
2answers
33 views

check my solution to indefinite integral problem with arccos

So we had homework it asked us to find $$\int\arccos(x)dx$$ I have found that $$\int\arccos(x)dx=x\arccos (x)+\sqrt{1-x^2}+c$$ Is this right?
3
votes
2answers
55 views

Simple integral (involving trig)?

This seems like a simple problem, but my trig manipulations are leading to a dead end. Compute: $$\int\frac{\sin^2(x)}{1 - \tan(x)} dx$$ Working thus far: Replace $$\tan(x) = ...
0
votes
1answer
28 views

Find Indefinite of root function

I don't know how to find this strange integral $\int{\sqrt{\dfrac{x-4}{x+2}}\dfrac{dx}{x+2}}$ Please help me solve this problem
0
votes
2answers
71 views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
3
votes
3answers
112 views

How do I integrate $\frac{1}{x^6+1}$

My technique so far was substitution with the intent of getting to a sum of three fractions with squares in their denominators. $t = x^2 \\ \frac{1}{x^6 + 1} = \frac{1}{t^3+1} = ...
1
vote
1answer
80 views

How to evaluate the integral : $I=\int\frac{2-x+(x-1)\ln x-\ln^2x}{(1+x\ln x)^2}dx$

How to evaluate the integral: $$I=\int\frac{2-x+(x-1)\ln x-\ln^2x}{(1+x \ln x)^2}dx.$$ Help me, thanks :/
2
votes
1answer
67 views

Integration double angle

How should i simplify this before applying integration. Have tried the $1-\cos2x=2\sin^2x$ but am still stuck on solving it $$\int\left(\dfrac{\cos2x}{1-\cos4x}\right)dx$$
0
votes
2answers
88 views

Integral of $\ln^2(x^2-1)/x^4$

I need to solve the following indefinite integral: $$\int \frac{\log^2(x^2-1)}{x^4}dx.$$ ($\log$ is the natural log) It's a past paper question from my uni exam so I don't think the answer is as ...
0
votes
2answers
77 views

I cannot find the following integral in an integral table.

In the appendix A of this paper there is an integral that the author says can be solved using any good integral table. However I cannot seem to find it on any integral table (ex: gradshteyn and ...
1
vote
3answers
75 views

Is the definite integral of the function necessarily the anti-derivative?

Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$ By the integral definition, g(x) is the area under the curve of f(x) from 1 to x. eg: g(5) is the area under f(x) from 1 to 5. I ...
2
votes
2answers
52 views

Integrate $\int\frac{5x-7}{x^2-3x+2}$

I want to integrate $\int\frac{5x-7}{x^2-3x+2}$ but my result differs from the one on Wolframalpha http://www.wolframalpha.com/input/?i=integrate+%285x-7%29%2F%28x%5E2-3x%2B2%29 I did the following ...
0
votes
1answer
44 views

Indefinite Integral Question - What kind of substitution?

I've been trying to solve this integral for the past two hours, but haven't gotten anywhere: $$ \int \frac {dx}{2\sqrt{x-4}+x} $$ I've tried various kinds of substitutions to no avail. Even just ...
4
votes
2answers
97 views

Evaluate $\int\frac{\sqrt {25 - x^2}}{ x^4}$

I'm pretty sure the method used is trig substitution. But I'm having trouble setting up and solving the problem.
3
votes
4answers
91 views

Evaluate the integral $(x-2) e^x$

I think this problem can be solved using integration by parts. So I set it up as $u = e^x$ and $du = e^x, dv = (x-2)$ and x = (x^2)/(2)-(2x) But I don't think I'm getting the right answer. Are my $u$ ...
4
votes
2answers
144 views

How to integrate this integral

I am having trouble solving this (note: we have not studied it yet nor was Google of any help) $$\int e^{x^x}\, dx$$
5
votes
6answers
306 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
1
vote
3answers
103 views

Help with integral $\int\frac{1}{\sqrt{\tan x}}dx$

I tried to solve by parts but it did not help.
1
vote
2answers
49 views

Help with integral with $\arcsin x$.

$$\int \frac{(1+x^2)\arcsin x}{x^2\sqrt{1-x^2}}dx$$ I saw that $$(\arcsin x)'=\frac{1}{\sqrt{1-x^2}}$$ and I tried to solve it "by parts"
0
votes
3answers
100 views

Evaluating $\int \frac{\operatorname d \! x}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$

How do you integrate $$\frac{1}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$$ or simply $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
2
votes
2answers
113 views

evaluation of $\int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$

Evaluation of $\displaystyle \int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$ $\bf{My\; Try::}$ First we will convert $\displaystyle \frac{\cos x+\sin x}{\cos x-\sin x} = ...
2
votes
1answer
88 views

How would I integrate $e^{e^x}$?

Is there a way to integrate: $e^{e^x}$ without using a Taylor or McLaurin Series expansion?
2
votes
3answers
80 views

Find the integral : $\int\frac{dx}{x^\frac{1}{2}+x^\frac{1}{3}}$

Find the integral : $\int\dfrac{dx}{x^\frac{1}{2}+x^\frac{1}{3}}$ Please guide which substitution fits in this I am not getting any clue on this .. thanks..
4
votes
2answers
132 views

Help solving an integral.

$$\int \frac{\sqrt{t+2}}{e^t}\,dt$$ I have tried integration by parts, but that is leading me no where. I typed it into Wolfram Alpha, but don't know much about erf function, just know what ...
0
votes
1answer
13 views

I need to find know how to integrate $x$ multiplied by a function to a power that is a fraction.

I know how to find integral functions normally, but when I try to find it from say $x\sqrt{4-x^2}$, I get completely lost.This screws me up in both indefinite and definite integration, so please help
1
vote
1answer
57 views

Evaluating $\int \cos^{-1}\left(\frac{x^2+a^2}{x^2-b^2}\right)x^2 dx$

How to evaluate the integral $$\int \cos^{-1}\left(\frac{x^2+a^2}{x^2-b^2}\right)x^2 dx$$$a<b.$ I posted a similar question here. Thanks in advance.
3
votes
3answers
103 views

Series Expansion Of An Integral.

I want to find the first 6 terms for the series expansion of this integral: $$\int x^x~dx$$ My idea was to let: $$x^x=e^{x\ln x}$$ From that we have: $$\int e^{x\ln x}~dx$$ The series expansion of ...
-2
votes
3answers
88 views

Evalute This Integral Function Using Trigonometric Function?

Evaluate$$\int x\sqrt{x^2 - 4}\,dx$$using trigonometric functions.
7
votes
4answers
760 views

Why should we get rid of indefinite integration?

It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that. ...
2
votes
1answer
107 views

The constant of integration during integration by parts

When integrating by parts, at what point does the constant come in? The rule has always been like this: $$ \int u\,dv = uv - \int v\,du $$ The explanation is that this the "reversal" of the product ...
1
vote
2answers
82 views

How can I integrate $\frac{\ln x - 1}{(\ln x)^2}$?

I've been stuggling with integrating this: $$ \frac{\ln x - 1}{(\ln x)^2} $$ Could you help? My guess is integration by parts but can't figure out how. Thanks!
1
vote
2answers
72 views

Integration of $\frac{1}{\log(x)}$

Please integrate $\frac{1}{\log(x)}$ and illustrate the steps of your method of integration. I have already tried integration by parts following the ILATE rule and otherwise. Eventually it forms a ...
1
vote
2answers
120 views

Integral of $\frac{1}{x\ln(x+1)}$

I'm trying to get my head around calculating $$ \int\frac{1}{x\ln(x+1)}dx. $$ I can't seem to get anywhere. I tried parts and substitutions, but that $(x+1)$ is always in the way. Any suggestions? ...
1
vote
5answers
69 views

Integral by parts of this function

I have this integral to solve: $\int t \sqrt{1+t}\,dt$ The book says to solve it by part but i have no clue about how to do so. Can someone help me?