7
votes
4answers
82 views

Integral of $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $dz=1/\sqrt{1+x^2}dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb v}dz=x\int zdz-\int ...
0
votes
1answer
59 views

Integrals related to the function $F(x) = \int_1^x (e^t/t )\, dt$

I'm having some trouble with part of a problem from Apostol Volume 1(Section 6.26, Number 6). For completeness I'll include the whole question: A function $F$ is defined by the following indefinite ...
4
votes
4answers
100 views

Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
0
votes
2answers
43 views

Calculus long division $\int\frac{y^4+3y^2-1}{y^3+3y}\ dy$

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part. $$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...
4
votes
4answers
65 views

Integration of some floor functions

Can anyone please answer the following questions ? 1) $\int$ $ \left \lfloor{x}\right \rfloor $ $dx$ 2) $\int$ $ \left \lfloor{\sin(x)}\right \rfloor $ $dx$ 3) $\int_0^2$ $\left ...
0
votes
1answer
90 views

Calculate the integral of $\sqrt{36\sin^2(2t)+6\cos^2(t)}$

During an arc length calculation I reached the following integral and I am having hard time calculating it: $$\int\sqrt{36\sin^2(2t)+6\cos^2(t)}\,dt=\sqrt{6}\int\cos t \sqrt{24\sin^2(t)+1}\;dt$$ ...
2
votes
3answers
102 views

About integrating $\sin^2 x$ by parts

This is about that old chestnut, $\newcommand{\d}{\mathrm{d}} \int \sin^2 x\,\d x$. OK, I know that ordinarily you're supposed to use the identity $\sin^2 x = (1 - \cos 2x)/2$ and integrating that ...
3
votes
3answers
445 views

How to find the antiderivative of $\frac{1}{x^2(1+x^2)}$?

How to find the antiderivative of $\dfrac{1}{x^2(1+x^2)}$? I recognized that this can be done with trigonometric substitution and I let $x = \tan(x)$ and ended up with $\dfrac{1}{\tan(x^2)}$; then I ...
0
votes
2answers
102 views

Evaluate $\int \frac{du}{(u^2+2)^2}$ [closed]

Someone can help me with some idea to solve the integrate $$\int \frac{du}{(u^2+2)^2}$$ I tried to solve it using trigonometric substitution, but it failed.
2
votes
2answers
130 views

How to evaluate the following indefinite integral? $\int e^{e^x}\mathrm dx$

I stumbled across this question: what's the value of the following integral? $$\int e^{e^x}\mathrm dx.$$ Furthermore I was required to demonstrate. On wolfram I got the result $\operatorname{Ei}(e^x)$ ...
1
vote
1answer
45 views

$\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx<\infty $ for some large $n$?

Fix $y\in \mathbb R.$ Define, $$I(y)=\int_{\mathbb R} \frac{1+x^{2}}{(1+|x-y|)^{n}} dx.$$ My Question is: Can we show that $I(y)<\infty$ for some large $n\in \mathbb N$ ? If yes, what is a value ...
1
vote
1answer
77 views

Tough integral with many radicals

I am completed baffled with this integral $$\int\left[\dfrac{1}{x^{1/3}+x^{1/4}}+\dfrac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\right]\mathrm dx$$ Any tips?
3
votes
4answers
118 views

Evaluate $\int{\frac{xe^x}{(1+x)^2} dx}$

How would I evaluate this integral? $$\int{\frac{xe^x}{(1+x)^2} dx}$$ I know I need to use parts but I ended up getting a very complicated expression to integrate the second time.
3
votes
3answers
62 views

How do I solve $\int\frac{7}{\sqrt{x}(x+4)}~\mathrm{d}x$?

I am trying to solve $\int\frac{7}{\sqrt{x}(x+4)}~\mathrm{d}x$. So far I have $$7\int\frac{1}{\sqrt{x}(x+4)}~\mathrm{d}x$$ $$u=\sqrt{x}$$$$\mathrm{d}u=\frac{1}{2\sqrt{x}}$$ and this is where I'm not ...
4
votes
7answers
150 views

How do I go from this $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$?

So I am doing $\int\frac{x^2-3}{x^2+1}dx$ and on wolfram alpha it says the first step is to do "long division" and goes from $\frac{x^2-3}{x^2+1}$ to $1-\frac{4}{x^2+1}$. That made the integral much ...
0
votes
0answers
67 views

Indefinite integral $\int\frac{e^x}{x(1+\log(x))}dx$

How to integrate this integral $$\int\frac{e^x}{x(1+\log(x))}dx$$ My attempt: I try some subtitutions, $e^x=u$$\hspace{0.2cm}$ and $\hspace{0.2cm}$$1+\log(x)=u$ but these are not helpful.Please help ...
2
votes
3answers
65 views

Trigonometric integral evaluation: $\int 4 \sin^4 x \cos^3 x \,dx$ [duplicate]

Evaluate the following integral $$\int 4 \sin^4 x \cos^3 x \,dx$$ I can do simple integration problems, but problems like this seem to stump me, I created this problem so I could solve and compare it ...
1
vote
3answers
52 views

I need help solving this indefinite integrals problem?

I am doing indefinite integrals homework, and this problem popped up. I hate to post on here without any personal insight on the problem, but I really have no idea on how to approach this.I do not ...
2
votes
1answer
79 views

Evaluate $\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx $

Evaluate the indefinite integral $$\int\frac {\csc^2{x}-2005}{\cos^{2005}{x}} dx$$ I tried multiplying and dividing by $\sec^2 {x} $ and then setting $\tan{x}=y$ but no good. Then I set $\cos ...
3
votes
4answers
133 views

How would I go about evaluating $\int \frac{x}{(9-8x^2)^3}dx$?

So I have homework on webAssign (a site used by my college), and I am not understanding the logic as to why I am taking the steps into solving the integral it is telling me to take. So I'll list the ...
0
votes
2answers
46 views

What is happening to the '2' in this integral?

It is the indefinite integral: $\int \frac{1}{2x-6}$ I am trying to understand it and looking the last step goes from $\frac12 \log(2(x-3))$ to $\frac12 \log(x-3)$ Can someone explain to me why the ...
0
votes
0answers
29 views

Integrating the logarithm of a function including a square root of a second degree polynomial

I have been trying for some time to calculate the following integral: $$\int \ln\left(k+\sqrt{ax^2+bx+c}\right)\ dx$$ where k, a, b and c are real numbers. I have tried several strategies, but without ...
2
votes
3answers
185 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
0
votes
0answers
57 views

Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
2
votes
4answers
105 views

Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
4
votes
2answers
142 views

Evaluation of $\int\frac{\sqrt{\cos 2x}}{\sin x}dx$

Evaluation of $$\displaystyle \int\frac{\sqrt{\cos 2x}}{\sin x}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{\sqrt{\cos 2x}}{\sin x}dx = \int\frac{\cos 2x}{\sin^2 x\sqrt{\cos 2x}}\sin xdx ...
5
votes
1answer
121 views

How can I evaluate this indefinite integral? $\int\frac{dx}{1+x^8}$

How do I find $\displaystyle\int\dfrac{dx}{1+x^8}$? My friend asked me to find $\displaystyle\int\dfrac{dx}{1+x^{2n}}$ for a positive integer $n$. But looking up I am getting pretty noisy answer for ...
2
votes
2answers
152 views

integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $

This is a tough one. Thanks. $$\int \frac {x dx}{\sqrt {1+x^{10}} } $$ This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried: 1) substituting u for x^5 to get ...
2
votes
4answers
136 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
2answers
51 views

Doubt in integral substitution

I am not able to figure out what substitution to use in the following integral $$ \int \frac{(x-1)e^x}{(x+1)^3}dx $$ Any help would be appreciated.
4
votes
5answers
84 views

Integration problem $\displaystyle \int \frac{dx}{x(x^3+8)}$

$$\int \frac{dx}{x(x^3+8)}$$ I think I'm supposed to use partial fractions, but I am unsure of how to start the problem. Any help would be appreciated.
1
vote
3answers
94 views

Evaluate $\int \frac{1}{(2x+1)\sqrt {x^2+7}}dx$

How to do this indefinite integral (anti-derivative)? $$I=\displaystyle\int \dfrac{1}{(2x+1)\sqrt {x^2+7}}dx$$ I tried doing some substitutions ($x^2+7=t^2$, $2x+1=t$, etc.) but it didn't work out.
0
votes
0answers
36 views

The negative integral meaning

Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, ...
-1
votes
1answer
39 views

Proving the indefinite integral $ \int \frac{1}{u^2(a+bu)}du $ [closed]

How can I prove that the indefinite integral $$ \int \frac{1}{u^2(a+bu)}du $$ is equal to $$ -\frac{1}{a}\left(\frac{1}{u}+\frac{b}{a}\ln\left|\frac{u}{a+bu}\right|\right)+C\ ? $$
1
vote
3answers
115 views

Evaluate $\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$

$$\int \frac{\tan^3x+\tan x}{\tan^3x+3 \tan^2x+2 \tan x+6} dx$$ My approaches so far has been using substitution with $\tan x = t$ and $\tan \frac x2 = t$ but the calculations has been harder than I ...
4
votes
1answer
82 views

Integrate : $\int(\sin x+\cos x)^ndx$

Problem : $$\int(\sin x+\cos x)^n\ dx$$ I am not getting any clue how to integrate this. Please help . I will be grateful to you. Thanks.
2
votes
1answer
49 views

Solving indefinite integrals gives multiple answers. Are all those answers correct?

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...
-1
votes
1answer
61 views

sin x integral qestions [duplicate]

How could the following integral be solved in a good manner? $$\int \frac{\sin(x)}{x}\;\mathrm{d}x$$ Regards:
1
vote
1answer
33 views

missing $j*\omega$ in integral

let us consider following integral according to property of delta function,we can write this intgeral as $\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$ or we can write as ...
6
votes
3answers
455 views

Are indefinite integrals unique up to the constant of integration?

We often write e.g. $$\int x^2 dx=\tfrac{1}{3}x^3+c$$ for any $c \in \mathbb{R}$, where $c$ is the constant of integration. We can show (via limits) that, if $g(x)=\frac{1}{3}x^3+c$, then ...
0
votes
1answer
61 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
2
votes
3answers
149 views

How to evaluate $\int \frac{x^6}{x^4-1} \, \mathrm{d}x.$

Evaluate the integral: $$\int \frac{x^6}{x^4-1} \, \mathrm{d}x$$ After a lot of help I have reached this point: $x^2 = Ax^3 - Ax + Bx^2 - B + Cx^3 + Cx^2 + Cx + C + Dx^3 - Dx^2 + Dx - D$ But now I ...
2
votes
1answer
60 views

Strange error concerning integration by parts

First, this is not homework; I just decided to try a classic integral in a non-standard way and came out with a strange result. The integral $I:=\int\frac{dx}{x\ln x}$ is well-known to equal $\ln\ln ...
8
votes
4answers
146 views

How to calculate $\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}\ dx$?

How to calculate $$\int\frac{1}{x + 1 + \sqrt{x^2 + 4x + 5}}dx?$$ I really don't know how to attack this integral. I tried $u=x^2 + 4x + 5$ but failed miserably. Help please.
4
votes
4answers
103 views

Evaluation of $ \int \tan x\cdot \sqrt{1+\sin x}dx$

Calculation of $\displaystyle \int \tan x\cdot \sqrt{1+\sin x}dx$ $\bf{My\; Try::}$ Let $\displaystyle (1+\sin x)= t^2\;,$ Then $\displaystyle \cos xdx = 2tdt\Rightarrow dx = ...
2
votes
2answers
27 views

how to calculate integrate about Heaviside

everyone,here I have a question about how to calculate $$\int e^t H(t) dt$$ where $H(t)$ is Heaviside step function thank you for your answering!!
1
vote
2answers
73 views

Indefinite integral of $x^x$

I've seen many many questions on the internet with answer that it cannot be done with elementary functions. Now I did this integration myself and got a pretty nice result. Since I've seen so many ...
4
votes
3answers
186 views

Integration by change the variable

Let, $\int_{-1}^1\sqrt{1+e^x}\operatorname{dx}$. Write as an integral of a rational function and compute it. Suggest: change the variable in order to eliminate the square root. My work was: ...
3
votes
5answers
123 views

Evaluate $\int \sqrt{1-x^2}\,dx$

I have a question to calculate the indefinite integral: $$\int \sqrt{1-x^2} dx $$ using trigonometric substitution. Using the substitution $ u=\sin x $ and $du =\cos x\,dx $, the integral becomes: ...
0
votes
1answer
54 views

Prove that $\iint_S \text{curl }\textbf{F} \cdot d\textbf{S} = 0$ where $S$ is a sphere.

Prove without using the divergence theorem. The proof using the divergence theorem is very obvious, but I need the proof which does not rely on the divergence theorem. Thanks in advance.