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 ...
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
0answers
64 views

Did I really fail the Gram Shmidt procedure, or the website had an error? linear algebra [closed]

below is the results from my website assignement. There is also my handwritten work-up to my answers. I don't see any error Ii made in finding "C", could the website have an error?
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} ...
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.
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 ...
-1
votes
1answer
29 views

condition for convergence to Riemann integral

Suppose that functions f and g are defined on [0, T] and bounded. Let [0,T] is divided into n point. that is , $t_1=0<t_1=T/n<...<T_n=T$ $\Sigma f(t_i)g(t_i)$ i want to show that as n ...
0
votes
2answers
56 views

Definite integral versus indefinite integral evaluation

Why evaluating $$\iint x\, \mathrm dx\, \mathrm dx$$ in $[0;2]$ is different from calculating $$\int^2_0 \int^2_0 x\, \mathrm dx\, \mathrm dx$$ ? What is the conceptual difference between the two?
0
votes
0answers
28 views

polar co ordinates integration

integrate the polar co ordinates $$ \int^{r=\infty}_{r=0} \int^{z=\infty}_{z=-\infty} \delta(r) \delta(z-z_s) dz dr$$ => I want to integrate the above equation. integral of $ \int^ {z=\infty} _ {z= ...
1
vote
0answers
36 views

Find the power series for a definite integral

I am a bit unsure when integration is used together with summation. Here is my question: Find power series for $\int_0^{1} \frac{\sin x}{x}dx$ in the form $\sum_{k=1}^{\infty} a_kx^k$ Here is what I ...
0
votes
2answers
93 views

fourier transform of sinc function

let us consider fourier transform of sinc function,as i know it is equal to rectangular function in frequency domain and i want to get it myself,i know there is a lot of material about this,but i want ...
1
vote
3answers
90 views

Is the definite integral of the function necessarily the anti-derivative?

Let's say you have a function defined as $$g(x)=\int_1^xf(t)dt$$ By the integral definition, g(x) is the area under the curve of f(x) from 1 to x. eg: g(5) is the area under f(x) from 1 to 5. I ...
5
votes
6answers
351 views

How to integrate $\displaystyle 1-e^{-1/x^2}$?

How to integrate $\displaystyle 1-e^{-1/x^2}$ ? as hint is given: $\displaystyle\int_{\mathbb R}e^{-x^2/2}=\sqrt{2\pi}$ If i substitute $u=\dfrac{1}{x}$, it doesn't bring anything: ...
0
votes
1answer
15 views

I need to find know how to integrate $x$ multiplied by a function to a power that is a fraction.

I know how to find integral functions normally, but when I try to find it from say $x\sqrt{4-x^2}$, I get completely lost.This screws me up in both indefinite and definite integration, so please help
7
votes
4answers
788 views

Why should we get rid of indefinite integration?

It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that. ...
1
vote
2answers
129 views

Integral of $\frac{1}{x\ln(x+1)}$

I'm trying to get my head around calculating $$ \int\frac{1}{x\ln(x+1)}dx. $$ I can't seem to get anywhere. I tried parts and substitutions, but that $(x+1)$ is always in the way. Any suggestions? ...
5
votes
1answer
85 views

Uses of integral calculus in discrete mathematics?

I have to do a project in my integral calculus class. But all the topics are too mainstream (parabolic arc calculation,archimedean approzimation of circle are,obtaining $E=mc^2\dots$ However I'm ...
2
votes
2answers
102 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
1
vote
0answers
22 views

Are all antiderivatives of a function an area or a (difference of areas) function of its derivative?

If you take $f(t)=2t$ then the integral $\int_a^xf(t)dt=x^2-a^2$ seems to be the general antiderative of $f$ and the notation of $\int f(t)dt$ makes sense as it is saying the antideritave of $f$ is ...
1
vote
2answers
102 views

Explain me definite integral…

So, as the derivative is a tangent of an angle between the $x$-axis and the corresponding tangent line, how can we represent an indefinite integral, and why is the area (definite integral) of, for ...
1
vote
2answers
277 views

The unbeatable $\int e^{1/\cos(x)} dx$ integral

Is there any way to express this in non-elementary functions? $$ \int e^{1/\cos(x)} dx$$ And/or to calculate this definite integrals? $$ \int_{-\pi/2}^{\pi/2} e^{1/\cos(x)} dx$$ $$ ...
0
votes
5answers
205 views

How to integrate $\int_{-\pi}^\pi|\cos(x)|dx$

I need to calculate the average value $\mu$ with the formula: $$ \mu = \frac{1}{b-a} \int_a^bf(x)\,dx $$ in my case: $$ \mu = \dfrac{1}{2\pi}\int_{-\pi}^\pi |\cos(x)|\,dx =\dfrac{1}{\pi}\int_{0}^\pi ...
4
votes
1answer
66 views

Find recursive forumula for integrals

I have to find recursve formulas for solving the following two integrals. The assignment tells one to find an Expression that leads from the calculation of $\dfrac{I_{2n}}{I_{2n+1}}$ to the ...
1
vote
5answers
107 views

Definite integration problem (trig).

I have this definite integral: $$ \int_0^\Pi \cos{x} \sqrt{\cos{x}+1} \, dx $$ For finding the indefinite integral, I have tried substitution, integration by parts, but I'm having trouble solving ...
3
votes
2answers
162 views

an intriguing integral $I=\int\limits_{0}^{4} \frac{dx}{4+2^x} $

I solved this integral: $$I=\int\limits_{0}^{4} \frac{dx}{4+2^x} $$ In a method similar to the one used in An interesting integral $I = \int\limits_{-1}^{1} \arctan(e^x)dx $. This integral ...
1
vote
1answer
404 views

Express the indefinite integral $\int\sin x^2~dx$ as a power series

What does this mean? I never saw this in my class/notes so I don't understand the conversion from integral to power series. Also if the integral were defined from $0$ to $1$, what new steps do I add? ...
0
votes
0answers
73 views

Indefinite integral with multiple values evaluated at endpoints to get definite integral

I am having trouble evaluating the definite integral $$\int_{-\pi}^{\pi} \frac{\sin\phi}{\pi\left( 1-\cos^2\Lambda\cos^2\phi \right)}d\Lambda$$ by the standard technique of the difference of value of ...
1
vote
1answer
90 views

Find point between two intervals which equals the average value.

Find the point in the interval [5,9] at which the function $f(x)=17e^{3x}$ equals its average value on that interval. So I've got my function as follows ...
0
votes
0answers
78 views

Integral $\int_{ \ 0}^{\ L} \exp{(\frac{-(x-x_0)^2}{4n})\sin({m\pi\over L}(x-A)}\ \mathrm dx$

How to find integral: $$I=\int_{ \ 0}^{\ L} \exp{\left(\frac{-(x-x_0)^2}{4n}\right)\sin\left({m\pi\over L}(x-A)\right)}dx$$ Thanks in advance. My try: By substitution I get $$z=(x-x_0)^2$$ $$\sqrt ...
1
vote
1answer
89 views

Integral of the Planck radiation formula

It's well known that the blackbody radiation law has ben derived by Plank and its mathematical formula is: $$W(\lambda,T)=\frac{C_1}{\lambda^5\left(\exp\frac{C_2}{\lambda T}-1\right)}$$ The definite ...
-1
votes
2answers
191 views

An arbitrary collection of integration exercises [closed]

1) I got the $A=2$ and $B=-1$. I think I'm just having trouble with the integration part now. Evaluate the integral $$\int_0^1 \frac{x-6}{x^2-6x+8}\,dx.$$ 2) For this one I got that $A=7$, ...
1
vote
4answers
70 views

Regarding u-substitution

1) $\displaystyle \int_{1}^{4} \frac{(\ln x)^3}{2x}dx$ 2) $\displaystyle \int_{}^{} \frac{\ln(\ln x)}{x \ln x}dx$ 3) $\displaystyle \int \frac{e^{\sqrt{r}}}{\sqrt{r}}dr$ 4) $\displaystyle \int ...
4
votes
1answer
198 views

Non elementary antiderivative of $\phi(\cos x,\sin x)$ when $\phi(x,y)$ is a rational real function?

With the method of Residues, we can calculate the integral \begin{equation}\int_{0}^{2\pi}\phi(\cos x,\sin x)\, dx \end{equation} where $\phi(x,y)=\frac{p(x,y)}{q(x,y)}$, ($p,q$ are polynomials of ...
2
votes
1answer
122 views

Help in understanding integration by changing the variable

I need help better understanding how, and why integration by changing the variable works (I've seen it's related to the derivative of a composite function $f(g(x))$), and generally tips and tricks, an ...
6
votes
3answers
2k views

What's the connection between the indefinite integral and the definite integral?

I want to understand the connection between the primitive function or antiderivative and the definite integral. My problem with this is the independent variable called t in the formula for the first ...