0
votes
2answers
39 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
23 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
175 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
55 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
96 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
118 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 ...
4
votes
1answer
117 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
147 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
124 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
43 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
83 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
92 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
37 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
109 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 ...
3
votes
1answer
77 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
47 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
59 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
32 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
450 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
59 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
115 views

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

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
59 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
143 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
92 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
70 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
182 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
111 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
53 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.
7
votes
2answers
146 views

How to evaluate the following integral? $\int \ln(e^x + c)~\mathrm dx$

I can't seem to find an answer for this kind of integration, and I'd like to know if there is an answer for it, and if yes what is it. $$\int \ln(e^x + {c})~\mathrm dx\,,$$ where $c$ is a constant. ...
1
vote
4answers
83 views

Integrating powers of linear and quadratic functions

How can I integrate function such as $(x+9)^3$? I obviously know that I can expand the function and integrate it normally. However, that is possible and feasible only as it is of third degree. What if ...
2
votes
6answers
124 views

How to evaluate the following indefinite integral? $\int\frac{1}{x(x^2-1)}dx.$

I need the step by step solution of this integral please help me! I can't solve it! $$\int\frac{1}{x(x^2-1)}dx.$$
0
votes
1answer
43 views

How do I solve this Integral using an infinite series

I'm supposed to use an infinite series to solve $$\int\frac{e^{2x}}{x}dx$$ How do I solve this? I know the answer already, I don't even have to use infinite series to solve this, however it was ...
1
vote
3answers
50 views

Initial Value Problem: $\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $

Initial Value Problem: $$\frac {dy}{dx}=\frac {xy\sin x}{y+1}, y(0)=1 $$ I know I'm supposed to separate the values and integrate. this is where I get stuck: $$y+\ln y = -x\cos x+\sin x+c$$ This ...
0
votes
0answers
20 views

Difficult Integral in functional basis

Let $$g(x)=\int f\prime(x)\left[\frac{4}{3}x^2+4x^3+(2x^2+4x^3)f(x)+6x^2f^2(x)+xf^3(x)\right]dx$$ express $g(x)$ in terms of $\{1,x,x^2,x^3,....\}$ and $\{f(x),f^2(x),f^3(x),...\}$. Is there a clever ...
0
votes
2answers
51 views

Need some help with this integral

$$ \int{ \frac{1}{(3t-1)(t+1)(t-2)}}{dt} $$ How many ways are there to solve this integral without using partial fractions? Thank you.
0
votes
1answer
38 views

$\int Q(b-cx) dx =?$

I am unable to understand the following integral $$ \int Q(b+cx) dx = \frac{1}{c}\left[(b+cx)Q(b+cx)-\frac{1}{\sqrt{2\pi}}exp\{-\frac{(b+cx)^2}{2}\}\right] .......(1) $$ where Q(x) is defined as $$ ...
3
votes
4answers
75 views

If $I_n =\int \cot^nx\ dx$ then $I_0 +I_1 +2(I_2+I_3+ \cdots I_8) +I_9+I_{10}= $?

If $\displaystyle I_n =\int \cot^nx\ dx$ then find : $I_0 +I_1 +2(I_2+I_3+ \cdots I_8) +I_9+I_{10} $ = ? My approach : $I_n = \displaystyle\int \cot^{n-2} \cot^2x dx$ $\Rightarrow I_n = ...
3
votes
2answers
247 views

Did I integrate this correctly?

The question was: $$\int 2x^2 (x^3-4)^6\ dx$$ My answer was $\dfrac{(x^3-4)^7}{7} + C$. If my answer is wrong please show me the correct method. The textbook doesn't have answers so I turn to my ...
3
votes
1answer
41 views

Reverse Chain Rule Intergration

The question states; determine the following using reverse chain rule; $$\int\sin x\cos^5x\, dx$$ Can you show me how to do this when you let $U =$ either $\sin$ or $\cos$?
4
votes
4answers
59 views

Integral can't find how to do it: $\int\frac{2\ln(x)}{x}dx$

I have to find this integral $$\int\frac{2\ln(x)}{x}dx$$ This is how I began: $$\int\frac{2\ln(x)}{x}dx=2\int\frac{\ln(x)}{x}dx$$ Then I tried substitution $e^u=x$ to get $u=\ln(x)\longrightarrow ...
3
votes
4answers
95 views

Evaluate $\int\cos(\ln x^2)dx$

$$\int\cos(\ln x^2)dx$$ I've learned substitution method, integration by parts, and some other basic methods for integration, but I have no idea how to start this question. How should I start this ...
0
votes
1answer
49 views

How can I calculate the integral?

How can I calculate the following integral?? $$\int \frac{1}{t}u'(t) dt$$ I thought that I could it as followed: $$\int \frac{1}{t}u'(t) dt=\frac{1}{t}u(t)+\int \frac{1}{t^2}u(t) dt$$ but I don't ...
2
votes
3answers
245 views

Stuck on Indefinite Integral

Please help me. I have been stuck on this for ages :( $$\int \frac{1}{13\cos x+ 12}\,\mathrm{d}x$$ I appreciate any and all help. Thank you.
0
votes
3answers
115 views

What are other unexpected results of integration?

I have integral of $\dfrac{1}{t^2 + 1}$ and integral of $\dfrac{t}{t^2 + 1}$ whose output is $\arctan(t)$ and $\dfrac12\ln(t^2 + 1)$ respectively. Are there any similar unexpected results when we ...
17
votes
10answers
2k views

What is the most efficient method to evaluate this indefinite integral?

$$\int x^5 e^x\,\mathrm{d}x$$ Is there another, more efficient way to solve this integral that is not integration by parts?
3
votes
3answers
52 views

Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$.

Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
1
vote
2answers
62 views

How wrong is it? - A “proof” of the FTC that I came up with in high school by hand-waving.

In high school calculus, I was first taught that the area under a curve $f(x)$ between $x=a$ and $x=b$ is given by: $$ A = \lim_{\delta x \rightarrow 0} \sum \limits_{a}^{b} f(x) \delta x $$ Then ...
4
votes
2answers
229 views

Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $

I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$