2
votes
3answers
168 views

Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
2
votes
4answers
475 views

How useful/useless is the indefinite integral

After having met yet another person confused by indefinite integrals today, I've finally decided to ask the community. Do you think it makes sense to teach indefinite integrals? My opinion is that ...
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.
0
votes
0answers
57 views

Is the following integration of possible?

How to solve the following problem? $\int x^4/(1-x^4)^{3⁄2}dx$ I have tried the substitution, x=sinz, but failed.
2
votes
4answers
95 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
117 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 ...
1
vote
2answers
70 views

How do I find this integral $\int\frac{dx}{2x^4+2x^2-1}$

How I evaluate the above integral? $$\displaystyle\int\dfrac{dx}{2x^4+2x^2-1}$$ I have unsuccessfully tried it more than once. Is there a small substitution that I am missing? And is there any ...
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
31 views

Evaluation of Indefinite Integral resulting in Hypergeometric Function

I am attempting to derive the result: $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$ First, I start off with the binomial expansion of the integrand to ...
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 ...
2
votes
2answers
56 views

Integration by Tables problem

$$\int \frac {dx} {x(x^8-256)}$$ I am supposed to use the formula $$\int \frac {dx} {x(ax+b)} = \frac1b\ln\left|\frac x {ax+b}\right|+C $$ to find the integral. I don't know how to start. Help is ...
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.
3
votes
1answer
48 views

Does $\int { y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$ have a closed form

I am trying to solve the following indefinite integral $$F_Y(y) = \int {y\cosh \left(\beta y^2\right)}J_0\left(\gamma y^2 \right) dy$$ Where $J_0$ is the Bessel function of the first kind. I tried ...
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
108 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 ...
2
votes
4answers
72 views

indefinite integral computation $dx/(e^{-x}-x)$

Hi i'm trying to carry out the following indefinite integral: $$\int \frac{1}{e^{-q} - q} \, dq$$ mathematica is not helping me, and i think it is not solvable by substitution method. any idea on ...
0
votes
0answers
57 views

Mathematica Integrate gives back the integrand

i'm trying to Integrate the following function: (q (1 + q) - E^-q Sinh[q])/(-q + Cosh[q] Sinh[q]) - ( 2 q Tanh[q])/(-q + Cosh[q] Sinh[q]) I already solved ...
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 \ ...
4
votes
1answer
63 views

Determine the indefinite integral $\int \frac{\sin x}{\sin 5x}dx$ [closed]

Find the value of the given integral $$\int \frac{\sin x}{\sin 5x}dx$$
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 ...
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.
0
votes
2answers
65 views

Improper integral with removable discontinuity

Integrate , for $ \alpha > 2 $ $ \int_0^{\infty}\!\frac{x-1}{x^\alpha-1}\, dx. $ I would be intertest for any replies or any comments
2
votes
2answers
126 views

Compute the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$ or $I=\int_{d}^1 y^{-a}(1−y)^{b-1} dy$

I need to calculate the indefinite integral $I=\int y^{-a}(1−y)^{b-1} dy$, where $a$, $b$ are REAL NUMBERS and $b>0$. (my goal is to determine the definite integral $I=\int_{d}^1 y^{-a}(1−y)^{b-1} ...
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
109 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! $$\int\frac{1}{x(x^2-1)}dx.$$
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
246 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
40 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
244 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?