Questions tagged [definite-integrals]
Questions about the evaluation of specific definite integrals.
15,138
questions
1
vote
2answers
26 views
area under the curve $1/(2x)$
How can I find the area under the curve of $y = 1 / (2x)$ in the intervals $0 <x, y <1?$
1
vote
1answer
38 views
Calculate $ \int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$
I'm trying to evaluate
$$ \int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$$
but I'm not sure the best way to proceed. I've been trying to factor the inner terms in to rational functions of ...
2
votes
3answers
37 views
Find the volume between surfaces
I'm trying to find the volume between the surface $x^2+y^2+z=1$ and $ z=x^2+(y-1)^2$ but nothing works for me.
I made the plot and it looks like this:
How could you start? Any recommendation?
0
votes
2answers
25 views
How to find a volume of an object enclosed with planes without any projection?
How to find a volume of an object enclosed with planes:
$$x^2+z^2=4,$$
$$x+y=2,$$
$$x+y=-2,$$
$$x-y=2,$$
$$x-y=-2$$
without any projection?
When I project this object I know it is a truncated ...
0
votes
0answers
46 views
Solution for two-variable integral?
I have the following antiderivative and associated definite integral (with $n,t$ positive integers with $m \le n$, and $k$ an integer $\ge n$):
$$\int \biggl(\frac{1}{\left(n-t^2\right)^2+1}\biggr)^k ...
0
votes
2answers
54 views
Find the area bounded with the curve $x=1+t-t^3, \ \ y=1-15t^2.$
Find the area bounded with the curve $$x=1+t-t^3, \ \ y=1-15t^2.$$
I drew its graph and looks like this:
It seems to be symmetric about $x=1$. So, the idea that came to me was to cut by half and ...
0
votes
2answers
43 views
$\text{Q Find the area enclosed by the curves “}y^2+x^2=9\text{” and "}\left|\left(x^2-y\left|x\right|\right)\right|=1$
$\text{Q Find the area enclosed by the curves "}y^2+x^2=9\text{" and "}\left|\left(x^2-y\left|x\right|\right)\right|=1\text{" which contains the origin}.$
I tried to plot the graph on desmos. and got ...
4
votes
1answer
54 views
Differentiation of an integral depending on a parameter
Let $f(t):=\int_0^{\pi/2} \arccos\frac{t-\tan^2x}{t+\tan^2x}\,dx$, for $0\leq t\leq 1$. I would like to differentiate $f$ with respect to $t$ by taking the partial of the integrand:
$$
f'(t)
= \int_0^...
0
votes
2answers
71 views
Convergence $ \int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$
I was trying to solve the following integral:
$$\int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$$
But when I plugged it in to any online calculator, It said it couldn't find the integral and that it might ...
2
votes
1answer
70 views
Integration question with limit
I'm trying to solve the following question and have some ideas (please see below) but having a hard time trying to "connect all the dots" and using all the given information.
let $f:\left[a,b\right]...
-1
votes
2answers
42 views
How to solve $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}(\cos x+\sin x)^n \; dx$?
Evaluate the following integral $$\int_{-\pi/4}^{\pi/4}(\cos x+\sin x)^n\ dx=?$$
such that $n\in \mathbb N$
my work:
i thought of using binomial expansion of $(\sin x+\cos x)^n$ but it gives $n+1$ ...
0
votes
1answer
34 views
GIF of the sum $\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$
I am asked to find the GIF (greatest integer function) of the sum:$$\sum_{i=1}^{1000}\frac{1}{i^{2/3}}$$
I am able to find the lower limit of the sum by using the fact that
$$\sum_{i=1}^{1000}\frac{1}...
1
vote
1answer
33 views
I would like to compute the integral [closed]
$\int_0^{2\pi}-e^{\cos(t)}[\sin(t)\cdot \cos(\sin(t))+\cos(t)\cdot \sin(\sin(t)]\,dt$
2
votes
1answer
38 views
Finding a closed form of an integral: $\int_0^k\ln(a\sin^2(x)+(a+b)\cos^2(x))dx$
I am trying to find a closed form for the following integral:
$$\int_0^k\ln(a\sin^2(x)+(a+b)\cos^2(x))dx$$
And I know that $a>0$, $b\ge0$ and $k=(\pi(1+n))/2$ where $n$ is a natural number.
How ...
0
votes
2answers
31 views
Finding area bounded by 3 function that aren't constant
Find an area that are bounded by $3$ functions:
\begin{align}
š¦ &= š„ + 6
,\\
š¦ &= š„^3
,\\
2š¦ + š„ &= 0
.
\end{align}
I only found the solution if one of the functions is constant, ...
0
votes
0answers
22 views
Riemann sum to definite integral when the index is a sequence
Suppose $0\le c_n\le n$ and $c_n/n\to c\in(0,1)$ as $n\to\infty$. I am wondering if it is correct for an integrable function $f$ on (0,1) that
$$
\lim_{n\to\infty}\sum_{\sqrt{n}\le j\le n-c_n}f(\frac{...
5
votes
1answer
74 views
Integrate from $0$ to $2\pi$ with respect to $\theta$ the following $(\sin \theta +\cos \theta)^n$
$$\int_0^{2\pi} (\sin \theta +\cos\theta)^n d\theta$$
First I think about De Moivre's formula given by
$$(\cos x +i \sin x)^n=\cos (nx)+i\sin (nx)$$
I tried to apply it but I found myself lost !
Any ...
0
votes
1answer
27 views
Finding the area of an ellipse using change of coordinates
I would like to find the area of the ellipse $x^{2} +2xy +2y^{2} \leq 1$.
I was told to use the substitution $s = x+y$ and $ t=y$.
Using this, I found the Jacobian determinant to be $1$ and then ...
-1
votes
2answers
54 views
Why is $\frac{\int_{64}^{65}1.04^xdx}{\int_{24.5}^{25.5}1.04^xdx}=1.04^{39.5}\;?$ [closed]
I am not sure whether these two things are exactly equal or only approximately equal. Wolfram says the difference is zero. I also would like to know why they are equal (or approximately)
$$\frac{\...
2
votes
1answer
45 views
Does $\int_{x_1}^{x_2}\frac{1}{f^{-1}(x)}dx=\int_{f^{-1}(x_1)}^{f^{-1}(x_2)}\frac{f^{\prime}(y)}{y}dy$ hold?
Consider a differentiable function $f$ that is strictly increasing and positive. Numerical examples seem to show that:
\begin{equation}
\int_{x_1}^{x_2}\frac{1}{f^{-1}(x)}dx=\int_{f^{-1}(x_1)}^{f^{-1}...
3
votes
1answer
86 views
How to prove that $ \int_0^1 \frac{e^x}{x+1}dx\le \frac e2$ using the inner product.
Let
$$\langle f,g \rangle=\displaystyle\int_0^1f(t)g(t)dt$$ be an inner product over $C[0,1]$. How to prove that
$$\displaystyle\int_0^1 \dfrac{e^x}{x+1}dx\le \dfrac e2$$
My miserable work: by ...
0
votes
1answer
33 views
Integrating Dirac delta functions with trigonometric arguments
I am not sure about how to integrate Dirac delta functions which have trigonometric arguments. I am currently trying to work out $\int_{0}^{2\pi} \delta(\cos(\theta)-k)d\theta$, $\lvert k \rvert$ <...
6
votes
2answers
142 views
Angular integrals used in QED
I am reading a research paper
and am stuck at a point where the author uses angular integrals. I don't have any idea about it and would like help.
The angular integral is:
$$I_k (y)=\int_0^\pi {\...
0
votes
2answers
33 views
$ \text{why} -\int_{0}^{1}{({1 - t})^{n} - 1 \over t}d t = \int_{0}^{1}{t^{n} - 1 \over t - 1}d t ?$
Question link : Proving Binomial Identity without calculus
i have one doubt in the given answer below ,my doubts mark in red colour
My doubt is that $$ \text{why} -\int_{0}^{1}{({1 - t})^{n} - ...
0
votes
1answer
60 views
Evaluate: $\int_{0}^{2\pi}\frac{\cos(x)}{2+\cos(x)}dx$ [duplicate]
I'm trying to solve this integral by using complex analysis, but don't know clearly what to do next. Here is my approach:
$$\int_{0}^{2\pi}\frac{\cos(x)}{2+\cos(x)}dx = \int_{0}^{2\pi}\frac{\cos(z)}{...
-1
votes
1answer
46 views
Integral equation with linear term [closed]
I wanted to solve this integral equation with linear term , I'll be very grateful for explanation with step by step solution.
$y(t)=t+Ī±t\int_{0}^1dp py(p)$
1
vote
1answer
33 views
Evaluate: $\int_0^1 \frac{e^x(1+x) \sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $
Evaluate: $\int_0^1 \frac{e^x(1+x)\sin^2(x e^x)}{\sin^2(x e^x)+ \sin^2(e-x e^x)} \,dx $
My assumption put $t= x e^x $ then $\int_0^e \frac{\sin^2(t)}{\sin^2(t)+ \sin^2(e-t)} \,dt$ How can I proceed ...
0
votes
0answers
25 views
Solving the integral $\iint_G \frac{12}{yx^4}e^{1/xy} dA$ [closed]
I am having difficulties trying to determine the boundaries for the integral:$$\iint_G \frac{12}{yx^4}e^{1/xy} dA$$ with the region $$G=\{(x,y)\in \mathbb{R}^2 \backslash (0,0) | x^2 \leq y, 8x^2 \...
1
vote
1answer
45 views
What is the integral of the fractional part of a variable — related to integration by parts?
Let $\{x\}$ denote the fractional part of a variable, i.e. $\{x\}=x-\lfloor x\rfloor$. Would the integral of $\{x\}-\frac{1}{2}$ from $0$ to $1$ evaluate to $1$? That is
$$
\int_{0}^1 \{x\} -\frac{1}{...
0
votes
2answers
55 views
Using differentiation under integral sign, prove $\int_{0}^{\infty} e^{-(x^2+\frac {a^2}{x^2})b^2} dx=\frac {\sqrt {\pi}}{2b} \cdot e^{-2ab^2}$
Using differentiation under integral sign, prove that $\int_{0}^{\infty} e^{-\Big(x^2+\frac {a^2}{x^2}\Big)b^2} dx=\frac {\sqrt {\pi}}{2b} \cdot e^{-2ab^2}$.
My Attempt:
Let,
$$F(a)= \int_{0}^{\infty}...
2
votes
1answer
53 views
On a log-gamma definite integral
A very famous log-gamma integral due to Raabe is
$$\int_0^1 \log \Gamma (x) \, dx = \frac{1}{2} \log (2\pi).$$
Several proofs of this result can be found here.
I would like to known about the ...
0
votes
3answers
51 views
Integral of $\sin(3x-3a)\sin(x)$?
Is there any technique that allows us to solve integrals of the following form?
$$ \int_{0}^{a} \sin(3x-3a)\sin(3x)dx $$
0
votes
1answer
32 views
Riemann Integrable Function Problem
I have recently read a book on real analysis, more specific the part about Riemann Integrals. It says that if a function is bounded in a interval $[a,b]$, then it is integrable and also when a ...
1
vote
1answer
54 views
On supremum of a function which is defined in terms of an integral.
Let $s>0$, $a=\ln(1+tan(\pi/8))$, and we define $$f(s)=\int_{-\pi/2}^{\pi/2}\frac{2\pi e^{a+s\pi \cos \theta}}{2+2e^{a+s\pi \cos \theta}\cos(\pi \sin \theta)+e^{2a+2s\pi \cos \theta}} \sqrt{s^2\sin^...
1
vote
1answer
67 views
specify that the given statements are correct or not
If $\displaystyle\int_a^bf(x)dx>\displaystyle\int_a^bg(x)dx$ ⇒
i) g(x) ≥ f(x) for x ∈ [a,b]
ii) f(x) ≥ g(x) for x ∈ [a,b]
i thought that we can take ...
1
vote
1answer
54 views
Show that for every m and n value, $\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$
Show that for every m and n value, $$\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$$
I have no idea how to solve a question like that. Do I have to solve both parts of the equation and show ...
1
vote
2answers
43 views
Help with calculating the integral $\int_{-\pi}^\pi \cos\left(x/2\right) \cdot e^{ix} dx$ by using Eulers formula
I have to determine the following integral
$$
\int_{-\pi}^\pi \cos\left(x/2\right) \cdot e^{ix} dx
$$
by using Eulers formula
$$
\cos\left(x/2\right) = \frac{e^{ix/2}+e^{-ix/2}}{2}
$$
we have that
\...
-2
votes
1answer
48 views
Conditions for a Riemann integral $\int\limits_{a}^{b}f(x)dx$ to be finite
For a Riemann integral $\int\limits_{a}^{b}f(x)dx$ to be finite is it necessary for the integrand to be finite both at the lower and upper limits of integration? Here, I would like the specifically ...
0
votes
1answer
33 views
Integrating a non-function - Work Done using P-V graph
The above figure was given to us, with the question saying
An ideal gas is taken from state $A$ to state $B$ via the above three processes. Then for heat absorbed by the gas, which of the ...
2
votes
1answer
58 views
Integral of a Gaussian times the square root of a quadratic
The integral:
I am curious if the following integral can be computed analytically.
$$J=\int_{-\infty}^{\infty}e^{-a(x-b)^2}\sqrt{x^2+c^2}dx,\hspace{1cm}a,b,c\in\mathbb{R}^{+}$$
What I've tried:
[1]...
4
votes
4answers
74 views
Compute the integral $\int\limits_{0}^{\pi/2} \frac{dx}{\sqrt{1 + \sin x}}$
The original task was to find the arc length of $y = \ln(1 + \sin(x))$ where $x \in [0, \pi/2]$. Using the general formula for arc length of $y = f(x)$ I've got $\sqrt2 \int\limits_0^{\pi/2}\frac{dx}{\...
2
votes
1answer
33 views
Comparing summation and integration for non monotonic function
$$P=\sum_{r=3n}^{4n-1} \frac{r^2+13n^2-7rn}{n^3}$$.
$$Q=\sum_{r=3n+1}^{4n} \frac{r^2+13n^2-7rn}{n^3}$$.
$$I=\int_{3}^{4} (x^2-7x+13) dx = \frac{5}{6}$$
Compare the values of $P,Q,I$
I know ...
0
votes
3answers
38 views
Need help to solve this integral. Don't know how to do [closed]
I need help to evaluate$$\int_1^2\frac{x^3+4}{x^2+2x}dx$$
0
votes
2answers
41 views
How can I calculate this integral considering the point where denominator is zero?
$$I=\int_{0}^{3}\frac{1}{(y-1)^\frac{2}{3}}dy$$
If I substitute $u=y-1$ and split this integral up and calculate it for $0ā¤y<1$ and $1<yā¤3$ then I get the anwer $I=3(2^\frac{1}{3}+1)$. But then ...
0
votes
1answer
21 views
Integration limits at points where denominator is zero
If the domintator of some function $f$ from $[a,b]$ to $ā^2$, be equal to zero at some point $cā[a,b]$, does it necessarily imply that the function is not integrable on $[a,b]$? If yes, how should the ...
3
votes
0answers
89 views
Evaluate $\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$ [duplicate]
Evaluate $$I=\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$$
We have $$\frac{\ln(1-x)}{x}=-\sum_{k=1}^{\infty}\frac{x^{k-1}}{k}$$
Hence $$I=-\sum_{k=1}^{\infty}\left(\frac{1}{k}\int_{0}^{1}x^{k-1}\...
5
votes
3answers
220 views
How to solve this integral with transformation to polar coordinates?
How do I determine new limits when transforming to polar coordinates. I have this example, and I don't know how to solve it correctly.
$$
\iint_D \frac{\ln\left(x^2+y^2\right)}{x^2+y^2}\,dx\,dy
$$
...
1
vote
2answers
115 views
Ahmed integral revisited $\int_0^1 \frac{\tan ^{-1}\left(\sqrt{x^2+4}\right)}{\left(x^2+2\right) \sqrt{x^2+4}} \, dx$
How can we prove (it is numerically verified already):
$$\int_0^1 \frac{\tan ^{-1}\left(\sqrt{x^2+4}\right)}{\left(x^2+2\right) \sqrt{x^2+4}} dx=-\frac{5 \pi ^2}{16}-\frac{1}{4} \tan ^{-1}\left(\sqrt{...
0
votes
0answers
16 views
Definite integration of shape with square cross sections perpendicular to y-axis. [closed]
(Sorry for the number of decimal places I'm just doing some calculations that require great precision.) The base of a solid is the region enclosed by the curve y=0.09399x^(2) -17.77171x +870.15519 and ...
0
votes
2answers
63 views
Integration that looks like fourier transform [closed]
There is a second order integral which looks like a fourier transform as shown below.
$$
\int_{-a/2}^{a/2}
\int_{-a}^{a}
\left(1-\frac{|x|}{a}\right)
e^{jkx \sin\theta \cos\phi} e^{jky \sin\...
