Question about finding the primitives of a given function, whether or not elementary.

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-2
votes
1answer
14 views

Questions on Indefinite Integration 2

Sorry to bother you guys, I have difficulty starting these questions. I have the answers, but I just can't seem to start the questions off. Really appreciate your efforts in helping me out. Thank you ...
0
votes
4answers
22 views

Questions on Indefinite Integration

Hi would like to know who yall will go about integrating this function. I have tried by substituting with u=2(x-1)^0.5 for question 1
0
votes
2answers
24 views

Evaluate the indefinite integral $\int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $ (using substitution) The answer is: $\frac {1}{24} ...
2
votes
3answers
32 views

Evaluate $\int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $ (using substitution) The answer is: $\frac{2}{11} \sqrt{sec(11 x)} + C$ I don't get ...
2
votes
1answer
25 views

Nature of an improper integral

I want to study the convergence of this integral at 0: $$ \int_0^{1}\frac{e^{\frac 1 t}}{\sqrt{t(1+t^2)}}\;dt. $$
1
vote
2answers
45 views

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$

Find the integral: $\int x^{7/2} sec^2(2+x^{9/2}) \mathrm{d}x$ Can I multiply and distribute the $ \ x^{7/2}\ $ and $ \ sec^2 \ $ together. What is the strategy to solve this problem.
1
vote
3answers
42 views

Evaluate the integral $\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x$

Evaluate the integral $$\int \frac{x}{(x^2 + 4)^5} \mathrm{d}x.$$ If I transfer $(x^2 + 4)^5$ to the numerator, how do I integrate?
2
votes
3answers
48 views

Find the integral: $\int ( 4x -1 +3 \sqrt{x})\mathrm{d}x$

I have to find the following integral: $$\int ( 4x -1 +3 \sqrt{x})\mathrm{d}x$$ My answer is $2x^2 -\ 1x + \frac{2\sqrt{27}}{3}$. Am I right?
1
vote
0answers
37 views

Integrating a rational function of exponentials

Let $\gamma ,\mu > 0$ be positive real constants and $\beta \in \mathbb{R}$ be a real constant. How can I evaluate the following indefinite integral? $$ \int \frac{e^{2\gamma t} (e^{-\mu t} - ...
3
votes
0answers
87 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
16
votes
3answers
127 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
3
votes
3answers
69 views

Evaluation of $ \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos xdx $

Evaluate $$\displaystyle \int_{0}^{\frac{\pi}{4}}\left(\cos 2x \right)^{\frac{11}{2}}\cdot \cos x \,dx .$$ $\bf{My\; Try::}$ Let $$\displaystyle I = \int \left(\cos 2x \right)^{\frac{11}{2}}\cdot ...
-6
votes
2answers
44 views

Two indefinite integral problems. [on hold]

Please help me out in these $$\int \frac{dx}{1-3\sin (x)}.$$ Second $$\int \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}dx.$$
2
votes
2answers
55 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluation of Integral $\displaystyle \int\frac{1}{x^2\left(x^4+1\right)^{\frac{3}{4}}}dx$ $\bf{My\; Try::}$ Let $\displaystyle x = \frac{1}{t}\;,$ Then $\displaystyle dx = -\frac{1}{t^2}dt\;,$ ...
1
vote
2answers
65 views

Finding $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$

Find $\int \frac{1+\sin x \cos x}{1-5\sin^2 x}dx$ I used a bit of trig identities to get: $\int \frac {2+\sin (2x)}{-4+\cos(2x)}dx$ and using the substitution: $t= \tan (2x)$ I got to a long ...
0
votes
1answer
16 views

About Weierstrass / Tangent half-angle substitution

From: http://en.wikipedia.org/wiki/Tangent_half-angle_substitution How did $\frac 1 {2\cos ^2 \frac x 2}$ become: $\frac {1+t^2} 2$? From the substitution of $\cos x$, it should be similar to: ...
2
votes
1answer
48 views

Integrating $\int \frac{\sin^3x}{(\cos x)^\frac 4 3} dx$

Find: $\int \frac{\sin^3x}{(\cos x)^\frac 4 3} dx$ My attempt: Set $u=(\cos x)^\frac 4 3 $ so $du= \frac 4 3 (\cos x)^\frac 1 3 \sin x dx \Rightarrow dx= \frac 3 {4 (\cos x)^\frac 1 3 \sin ...
0
votes
1answer
35 views

Difficult integral $\frac{du}{u}=\left(\frac{x+y}{x}\right)dx$ in PDE

The linear problem is given as $$x\frac{\text{$\delta $u}\backslash }{\text{$\delta $x}}\text{+y}\frac{\text{$\delta $u}\backslash }{\text{$\delta $y}}\text{=(x+y)u}$$ with $u = 1$ on $x=1$ with ...
1
vote
2answers
73 views

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

I need to find $\int\frac{1}{x^2-x-1}dx$ and I don't know what to do. I've thought about substitution or partial fractions but neither has worked.
0
votes
1answer
31 views

First order differential equation (with a logistic function)

I came across this first order differential equation $$ f'(x) = \left( \frac{1}{x} + \frac{g'(x)}{g(x)} \right) f(x) - c \frac{g'(x)}{g(x)} \textrm{,}$$ where $g(x)$ is this logistic type function $$ ...
1
vote
2answers
61 views

Evaluating $ \int \frac{1}{5 + 3 \sin(x)} ~ \mathrm{d}{x} $.

What is the integral of: $\int \frac{1}{5+3\sin x}dx$ My attempt: Using: $\tan \frac x 2=t$, $\sin x = \frac {2t}{1+t^2}$, $dx=\frac {2dt}{1+t^2}$ we have: $\int \frac{1}{5+3\sin x}dx= 2\int ...
0
votes
0answers
27 views

Indefinite integral of du=(x^2+y^2)dx/y solution check

Is the solution to the ODE $$\text{$\int $du=$\int $}\frac{\left(x^2+y^2\right)}{y}\text{dx}$$ equals to $$u=c+\frac{x^3}{3 y}+\text{yx}?$$
1
vote
1answer
31 views

Indefinite Integral of $(x\,dx-y\,dy)$

The indefinite integral of $x\,dx-y\,dy$ is $\frac{x^2-y^2}{2}$. But my book work the solution to be $x^2-y^2$. Who is correct?
0
votes
1answer
35 views

Evaluation of an integral.

$$I_1=\int \frac{a^2\sin^2(x)+b^2\cos^2(x)}{a^4\sin^2(x)+b^4\cos^2(x)}dx$$ I tried writing it as $$\int \frac{a^2+\cos^2(x)(b^2-a^2)}{a^4+\cos^2(x)(b^4-a^4)}dx$$ But I don't know how to proceed. ...
1
vote
1answer
75 views

How to integrate $\frac{1}{(x^2-4x+6)^{3/2}}$ with respect to x?

I need to find $\int\frac{1}{(x^2-4x+6)^{3/2}}dx$ but am not sure where to start. I tried breaking up the denominator into $(x^2-4x+6)$ and $(x^2-4x+6)^\frac1{2}$ but can't figure out how I would ...
3
votes
4answers
108 views

Integration: $\int\frac{1}{(x^2+x+1)^{1/2}} dx$

Find the value of $$\int\frac{1}{(x^2+x+1)^{1/2}} dx$$ Anyone can provide hint on how to integrate this, and how you know what method to use? (I mean, is there any general guideline to follow for ...
-2
votes
0answers
56 views

How do I evaluate the integral: $\int dz$ [on hold]

Given that $\int z~dz = \frac{1}{2}z^{~2}$ How do I evaluate the indefinite integral : $\int dz$, where $z=z_{0}e^\frac{\kappa t}{2}$ and $~z_0=const$
0
votes
3answers
64 views

How would I integrate $(x^2+1)^{\frac{3}{2}}$?

I need to find $\int(x^2+1)^{\frac{3}{2}}dx.$ I started by trying to split it into $\int (x^2+1)(x^2+1)^{\frac{1}{2}}dx$ and then integrating by parts but that didn't seem to be working out. Is there ...
3
votes
3answers
82 views

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

The task is to evaluate $$\int\frac{x^2}{\sqrt{1+x+x^2}}\,dx$$ My best approach has been substitution of $u=x+\frac{1}{2}$, and from there onto (some terrible) trig sub - finally arriving at a messy ...
-4
votes
0answers
20 views

Using a limit of a sum of an infinite number of rectangles, evaluate the integral. [closed]

Using a limit of a sum of an infinite number of rectangles, evaluate the integral. the integral $$\int_1^5(5-x^2)dx$$
-4
votes
0answers
30 views

Integration everywhere [duplicate]

Find this $\displaystyle\int\frac{e^ x}{x}\;dx$
2
votes
2answers
39 views

Integral by using substitution (How to proceed?)

Using the substitution $x=a\sin\theta$, or otherwise, find $\int\frac{1}{x^2\sqrt{a^2-x^2}}dx$. My attempt, $x=a\sin\theta$ $dx=a\cos (\theta)d\theta$. Then $\sqrt{a^2-x^2}=\sqrt{a^2-a^2\sin ...
0
votes
0answers
45 views

Integral of the reciprocal of the natural logarithm [closed]

What is the value of this integral $$\displaystyle\int\frac{1}{\ln(x)}\;dx$$
2
votes
3answers
155 views

How to solve this integral by a simple way?

I'm given $$\int \frac{x^3}{\sqrt{x^4+x^2+1}}dx$$ My attempt, Let $u=x^2$, $du=2xdx$ $$=\frac{1}{2}\int \frac{u}{\sqrt{u^2+u+1}}du = \frac{1}{2}\int ...
0
votes
0answers
17 views

Predicting equality/inequality of integrals of multivariable functions

Is it possible to predict equality/inequality, of indefinite integrals of multivariable fucntions, over a domain from equality/inequality respectively of those functions over the same domain? Does ...
2
votes
4answers
106 views

solving the integral of $e^{x^2}$

How to solve following integral? Any hints for the above integral ? $$\int{e^{x^2}} dx =?$$ I use change of variable $t=x^{2}$. so, $$\frac{1}{2}\int{\frac{e^{t}}{t^{\frac{1}{2}}}dt}$$ But I ...
1
vote
0answers
43 views

Is there a closed form expression for the following definite integrals?

I am looking for a closed form for these two integrals $$\int_{-\infty}^{-a}\text{d}x \frac{1}{|x|}e^{-\frac{1}{2}x^2\sigma^2}e^{i k |x|}+\int_a^{\infty}\text{d}x ...
0
votes
4answers
67 views

Evaluating $\int_{-a}^{a}\sqrt{a^2-x^2}dx$

Question: How to evaluate $$\int_{-a}^{a} \sqrt{a^2 - x^2} dx$$ This came up while trying to prove that the area of an ellipse is give by $\pi a b$ where $a$ and $b $are the major and ...
0
votes
3answers
71 views

Problem in indefinite integral. (Exponential)

I'm given this integral to integrate. I've no idea where to start with. Perhaps someone can give me some hints or guide me. Thanks a lot. $$\int\frac{(x^3)e^{x^2}{}}{x^2+1}dx$$
4
votes
5answers
87 views

Something wrong at $\int \frac{x^2}{x^2+2x+1}dx$

I have to calculate $$\int \frac{x^2}{x^2+2x+1}dx$$ and I obtain: $$\int \frac{x^2}{x^2+2x+1}dx=\frac{-x^2}{x+1}+2\left(x-\log\left(x+1\right)\right)$$ but I verify on wolfram and this is equal with: ...
2
votes
1answer
58 views

Arithmetic mean of $L^2$ function is $L^2$

I have found the following problem, to which I do not find the solution: Consider $f(x), x > 0$ a function such as $$ \int_0^\infty f^2(x) dx < \infty $$ and let $g(x) = \frac 1x \int_0^x ...
0
votes
0answers
28 views

Double Integral with integrand similar to bivariate normal density

I got a double integral like the following, $$\int_{0}^{\infty} \int_{y}^{\infty} xe^{-\frac{(x-by-c)^2}{2a}}ye^{-\frac{(y-e)^2}{2d}}dxdy,$$ where $a$,$b$,$c$,$d$,$e$ are viewed as some other ...
1
vote
2answers
146 views

Is this integral impossible to solve?

Is possible to express the antiderivative $$\int\frac{-3e^{-x^3}}{x^2}dx$$ in terms of elementary functions?
4
votes
2answers
56 views

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

I couldn't follow a step while reading this answer. Since I do not have enough reputation to post this as a comment, I'm asking a question instead. The answer uses "partial integration" to write this ...
2
votes
3answers
175 views

Where did I go wrong when doing this integral?

This is my integral $$ \int \frac{ (2x-3)}{(x^3 +10x)}\cdot dx \\ $$ This is my work $$ \int\frac{2x}{x^3 +10x}\cdot dx-\int\frac{3}{x^3 +10x}\cdot dx\\ $$ Looking at them separately: $$ ...
6
votes
3answers
92 views

How to integrate $(e^x + 2x)^2$?

I need to integrate $\int(e^x+2x)^2dx.$ I tried breaking it into $\int(e^x+2x)(e^x+2x)dx$ and then integrating by parts, but got stuck at $$ \int (e^x + 2x)^2\,dx = ...
3
votes
1answer
68 views

Evaluating: $I_1 = \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) $

$$I_1 =\int \sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) dx= ?$$ I tried substitution: $\sin^{-1} \left(\sqrt{\frac{x}{x+a}}\;\right) = \Xi$, but then I'm not able to do anything after the resulting ...
1
vote
1answer
27 views

calculate integral of given function

let us consider following integral while if we calculate from -infinity to plus infinity then it says that generally it should be 1/infinity +1/infinity right? which should be equal to ...
1
vote
0answers
27 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
0
votes
1answer
71 views

Can anyone please help with this integral. Very much appreciated..

Now , i've tried a couple of different substitutions and integrating partially but unfortunately to no luck, was wondering on your thoughts on it. I'd also be very thankful if someone were to have a ...