0
votes
0answers
49 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.
1
vote
4answers
82 views

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

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
2
votes
2answers
87 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
68 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
113 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
29 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
141 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
47 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
88 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
35 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
76 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
114 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
141 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
64 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
108 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
82 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
49 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
49 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
245 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
57 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?
0
votes
1answer
47 views

Integral with logarithm in denominator

I am having hard time to find following integral: \begin{equation} \int\frac{2x-1}{2x(1-x)}\cdot\frac{1}{2(1-x)+\log{x}}dx \end{equation} Can anyone help? Thank you.
3
votes
3answers
51 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 ...