0
votes
3answers
21 views

Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
3
votes
0answers
24 views

Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ This could be used to evaluate partial sums using knowledge of the ...
1
vote
1answer
35 views

Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
0
votes
0answers
45 views

Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
1
vote
3answers
44 views

Integration by parts: $\int e^{-\theta}\cos7\theta \;d\theta$

$$\int e^{-\theta}\cos7\theta \;d\theta$$ I started off by using $u=\cos 7\theta$ and$ \;dv=e^{-\theta}d\theta$, however, this just led me in a circle. I am now at: $$u=e^{-\theta},\;dv=\cos 7\theta ...
0
votes
2answers
31 views

Calculus 2: Strategy for Integration, Integral of e^(x+e^x)dx

How would you find $\int e^{x+e^x}dx$? I know I need to use $u$-substitution but I tried changing what I use for $u$ but I still could not get the right answer. If someone could push me in the ...
0
votes
1answer
42 views

How to find $F(x) = \int_x^{x^2} (2+\sqrt t )\, dt$ ?

I have this problem: $$ F(x) = \int_x^{x^2} (2+\sqrt t )\, dt $$ I have to solve the integral. I got $2x^2+\frac{2x^3}{3}-2x-\frac{2x^{3/2}}{3}$ However, I don't think that it correct.
2
votes
3answers
234 views

Why we use dummy variables in integral?

I want to know why we use dummy variables in integral? thanks so much.
0
votes
3answers
57 views

Evaluate the integral $\int_0^{1/4}\frac{x-1}{\sqrt{x}-1}\mathrm dx$

so I have this Integral I have to solve without a calculator. $$\int_0^{1/4}\dfrac{x-1}{\sqrt{x}-1}\mathrm dx.$$ How would I go about finding the antiderivative of that fraction?
-2
votes
1answer
29 views

Evaluating an integral with unspecified functions $f,g$, given other integrals with these functions

Suppose that $$\int_6^8(3f(x)-x)\,\mathrm dx=6$$ and $$\int_8^6(2x+4g(x))\,\mathrm dx=-8$$ Evaluate $$\int_8^6 (f(x)-5g(x))\,\mathrm dx$$ I have a problem. So, this one question asks me ...
1
vote
0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
2
votes
1answer
49 views

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$

If $\int \dfrac{f(x)}{x^2(x+1)^3}\hspace{1mm}dx$ is a rational function, and $f$ is quadratic function, such that $f(0)=1$. Then Find $f'(0)$ This looks like an interesting problem with an elegant ...
1
vote
3answers
35 views

integrate $\int e^{-iwt}dt$

I have this integral: $$ \int e^{-iwt}dt$$ I know that $\int e^{kx}=\frac{e^{kx}}{k}$ so therefore the $ \int e^{-iwt}dt$ would be $\frac{e^{-iwt}}{-iw}$ but Wolfram Alpha says that it is $\int ...
7
votes
0answers
93 views

Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$

What tools would you recommend me for evaluating this integral? $$\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \ dx$$ My first thought was to use beta function, but it's hard to get ...
0
votes
1answer
71 views

Finding total work by integration

The following tank is completely filled with water. Find the total amount of work done in pumping water out of the outlet. Note that the density of water is 1000 kg/m$^3$ I feel like I am ...
3
votes
1answer
40 views

Integral of [(1+2y^2)/(3-y)]dy (obtained from a differential equation)

This question actually arises from this Differential Equations question: Find the family of solutions for: $\displaystyle(1+2y^2)\frac{dy}{dx} + (3-y)\cos x = 0$ I ruled out the methods I've so far ...
1
vote
2answers
45 views

Integration of $1/(x^2+x\sqrt{x})$

The question is to evaluate $\displaystyle7\int\frac{dx}{x^2+x\sqrt{x}}$. My solution is attached. The problem of my solution is if I use partial fraction, loop will be made, and this makes ...
-2
votes
2answers
44 views

Integration of $(5x^2+2x-5)/(x^3-x)$

The problem is to evaluate $\int \frac{5x^2+2x-5}{x^3-x}\,dx$. This is the solution that I tried: I really have no idea of this problem. After check my solution, if there are any problem that ...
4
votes
0answers
61 views

Closed form for $\int_1^\infty\frac{dx}{\Gamma(x)}$

Is a closed form for $$\int\limits_1^{+\infty}\frac{dx}{\Gamma(x)}$$known? I tried to find it, but all well-known integrals involving gamma-function (such as of $\log\Gamma(x)$ or the like) don't ...
0
votes
6answers
83 views

For polynomials $f,g$, why is $\int_0^\infty \frac{fg}{e^x}\, dx$ absolutely convergent?

Why does the integral $\displaystyle \int_0^\infty \frac{fg}{e^x}\, dx$ have to be convergent for all real polynomials $f$ and $g$? Can anybody give me a proof?
1
vote
2answers
76 views

What do we mean by derivative of a function? What does it tell? [duplicate]

Taking the derivative of any kind of function is easy but I don't know why we take the derivative? Like $f(x)=x^2$ has the derivative $2x$, so what does it mean? I don't know how to define ...
8
votes
2answers
119 views
+100

Closed form of $I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

I'm looking for a closed form of this integral. $$I = \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx ,$$ where $\operatorname{Li}_2$ is the dilogarithm function. A numerical ...
2
votes
3answers
50 views

Finding the indefinite integral of a root function

I'm stuck on a particular integral problem. The problem is stated as: $$\int x \sqrt{2x+1} dx$$ My working thus far: $$\int x \sqrt{2x+1} dx = \frac{1}{2}x^2\frac{2}{3}(2x+1)^\frac{3}{2}$$ ...
5
votes
0answers
88 views
+200

Closed form for integral $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

I'm looking for a closed form of this definite iterated integral. $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$ From Vladimir ...
1
vote
0answers
30 views

Integral equation solution

I have an integral equations of the form $ \int s R(s) =s f(s)-\int f(s)ds \tag 1$ Can we solve this integral equation for $f(s)$ interms of $s,R(s)$ ? Means $R(s)=\psi(s,R(s))$ (with out integral ...
0
votes
0answers
52 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
1
vote
0answers
25 views

Looking for advice with the following integral

I have the following integral to evaluate: $$ \frac{1}{f(t)}\int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. ...
1
vote
1answer
63 views

Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy $$
6
votes
2answers
107 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
2
votes
2answers
47 views

Where should I place the notorious '+c'?

Consider the following proof - $$I=\int \sin (\ln x)dx\\\iff I=\sin(\ln x)x-\int\frac{ \cos (\ln x) }{x}\cdot {x} dx \\\iff I=x\sin (\ln x)-\int\cos(\ln x)dx\\\iff I=x\sin(\ln x )-[x\cos(\ln ...
1
vote
3answers
38 views

finding an indefinite integral of a fraction

(a) Show that $\frac{4-3x}{(x+2)(x^2+1)}$ can be written in the form ${\frac{A}{x+2} + \frac{1-Bx}{x^2+1}}$ and find the constants $A$ and $B$. (b) Hence find ...
4
votes
1answer
82 views

Stuck on this intergral $\int^\frac{\pi}{3}_\frac{\pi}{4} \frac{\tan^2x}{x-\tan x} dx $ calculus I

$$\int^{\pi/3}_{\pi/4} \frac{\tan^2x}{x-\tan x} dx $$ this is that I have tried $$\int^{\pi/3}_{\pi/4} \frac{\frac{\sin^2x}{\cos^2 x}}{x-\frac{\sin x}{\cos x}} dx $$ $$\int^{\pi/3}_{\pi/4} ...
6
votes
6answers
491 views

Two methods to integrate?

Are both methods to solve this equation correct? $$\int \frac{x}{\sqrt{1 + 2x^2}} dx$$ Method One: $$u=2x^2$$ $$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}} du$$ ...
0
votes
1answer
23 views

Problem with simplifying before integration

Can someone explain to me how did the du = 6y^(-1/3)dy went into the last equation?
2
votes
3answers
38 views

Evaluate trig function integral

I was struggling to evaluate this integral: $$\int x\sin^2(4x)\;dx$$ Every time I try again I end up with a different answer, my most recent answer I came up with is $$-\frac1{12} x\cos^3(4x) + ...
-1
votes
1answer
11 views

Given a Riemann Integrable function f, calculate the values of A,B,C [on hold]

Given a Riemann Integrable function f, calculate the values of A,B,C Any help will be thankful. Thanks!
1
vote
1answer
33 views

Integral involving exponents

How do we integrate $\int e^{C_1\frac{u^2+1}{u^2-1}} \ du\tag 1$ I could not find a proper substitution to convert it to a normal available form so that I can get a closed form of integration. $C_1$ ...
3
votes
0answers
56 views

Wicked domain of integration in a triple integral

I am dealing with a domain of integration of the form: $\left(\frac{x-y}{x+y}\right)^2+\left(\frac{y-z}{y+z}\right)^2+\left(\frac{x-z}{x+z}\right)^2\leq k$ The region looks like this (for $k=0.2$): ...
0
votes
0answers
15 views

A proof problem about intergral equation's root

Several days ago,my junior asked me the following problem: Let $$F\left( w \right) = \frac{1}{T}\int_0^T {M{x_C}\left( t \right)\cos \left( {tw} \right)dt} - \frac{{\sin \left( {{T_s}w} ...
6
votes
1answer
71 views

Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form \begin{equation} \sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1) \end{equation} where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
0
votes
0answers
41 views

integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...
3
votes
0answers
47 views

Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int\tan^2\theta\frac{\sin^2(\sec\theta\tan\theta)}{\sec^2\theta}d\theta \tag1$? How do I convert this in to a ...
0
votes
1answer
68 views

Integration with quadratic square root

What could be the techniques we need to use to solve this integration $\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ? Main issue here is the term inside $\sin^2()$. Very ...
14
votes
1answer
175 views
+200

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
3
votes
1answer
56 views

Find the exact length of the curve $y=\frac 12 x^2- \frac 12 \ln(x)$

Find the exact length of the curve $y = \frac 12 x^2- \frac 12 \ln(x)$, for $2 \le x \le 4$. My attempt: \begin{align} L&= \int_2^4 \sqrt{1+\left[x-\frac 1{2x} \right]^2} \, dx \\ &= ...
1
vote
1answer
47 views

Why does this double integral give me different answers?

According to the following link: http://www.instructables.com/id/Change-of-Variables-of-Double-Integrals/?ALLSTEPS The double integral ultimately evaluates to 1.58362 after variable replacement. ...
1
vote
1answer
22 views

order of integrals with independent limits

I was wondering if the following is true assuming that the limits are independent (like constants) $$ \int_{\alpha}^{\beta} \int_{\gamma}^{\psi} {xy} dx dy = \int_{\gamma}^{\psi} ...
11
votes
2answers
135 views

Evaluating $\int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$

Good evening! I want to compute the integral $\displaystyle \int_{0}^{\pi/3}\ln^2 \left ( \sin x \right )\,dx$. However I find it extremely difficult. What I've tried is rewritting it as: ...
1
vote
0answers
50 views

Find $\int \tan(\tan x)\hspace{1mm}dx$

Find $\int \tan(\tan x)\hspace{1mm}dx$ This is an Interesting problem, which I have been trying from different directions, nothing seems to work, its been a day on this one. Can anyone figure out ...
2
votes
4answers
56 views

Second order homogenous non-linear DE: $3(x')^2 - 2x''x=0$

How do I solve this for $x$? $$3\dot{x}^2-2\ddot{x}x=0$$ $$\Leftrightarrow$$ $$3(x')^2 - 2x''x=0 $$ Note: This comes from my working here(on stack exchange meta sandbox[newest activity]) List of ...