All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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3
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0answers
30 views

Solving integral $\int\frac{\sin x}{1+x\cos x}dx$

How I can find the anti-derivative? $$\int\frac{\sin x}{1+x\cos x}dx$$
3
votes
3answers
61 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
0
votes
0answers
12 views

Real analysis question involving inhomogenous linear ODE

So I had another problem like this but the ODE was homogenous, now there is a non zero right side. I completed part (i), $\large c(x) = \int \frac{b(x)}{g(x)} dx$. I am stuck on (ii) and the rest. ...
0
votes
1answer
25 views

Prove that there exists only one function f such that…

Prove that there exists only one function $$\big[f\in C\left ( \left [ 0,1 \right ],\mathbb{R} \right )s.t. f(x)=\frac{2}{5}\int_{0}^{1}(x^{2}+t^{5})f(t)dt+sin(x)\big] $$
4
votes
1answer
39 views

How to evaluate $\int_0^ \infty e^{-x\sinh(t)- \frac{1}{2} t}dt$?

$$ \int_0^ \infty e^{-x\sinh(t)- \frac{1}{2} t}dt $$ I tried doing it by parts and looking for differentials but I just keep getting back to the original expression. I can't think of a clever ...
1
vote
1answer
36 views

Prove by using step functions: $\int_{-b}^{b}\sin(x)\ dx = 0$

The Assignment: Let $b > 0$. Prove by using step functions: $$\int_{-b}^{b}\sin(x)\ dx = 0$$ The claim itself is obvious, but I have no idea how to prove it with step functions. My idea was ...
2
votes
1answer
21 views

Convergence Question:

This is related to the Dirichlet eta function. Does $$\int_1^\infty \frac{dx}{x^z}$$ converge for $Re(z)>1$? Just wondering. If so, then does $$\int_1^2 \frac{dx}{x^z}+\int_3^4 ...
0
votes
0answers
14 views

Finding surface integral of the paraboloid and disk

Let S be the surface consisting of the paraboloid $y=x^2 + z^2$ with $0 \leq y \leq 1$, and the disk $x^2 + y^2 \leq 1$. Let $S$ have an outward orientation. Compute the double integral of $\langle ...
0
votes
1answer
40 views

integral $I=\int_{-\infty}^\infty e^{-\alpha x^{2k}}dx$

$$ I=\int_{-\infty}^\infty e^{-\alpha x^{2k}} dx $$ The last problem was ill posed, and is answered in the post! You can disregard this post!
2
votes
0answers
31 views

Integral $\int_0^{\pi/3}\log\bigg( \frac{1+2\cos\theta}{2}+\sqrt{\left( \frac{1+2\cos\theta}{2} \right)^2-1}\ \bigg)d\theta.$

Hi I am trying to calculate this integral I given by $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
1
vote
1answer
27 views

Integrating an equation with both cos and tan

$$\int2\cos^5x\cdot\tan^6x\cdot dx$$ $$2\int\cos^5x\cdot\frac{\sin^6x}{\cos^6x}\cdot dx$$ $$2\int \frac{\sin^6x}{\cos{x}} dx$$ $$2\int\cos^{-2}x\cdot \sin^6x\cdot \cos{x}\cdot dx$$ ...
0
votes
0answers
20 views

Integration over rectangles

"Prove that a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is integrable over every rectangle in $\mathbb{R}^n $ if and only if it is integrable over every ball in $\mathbb{R}^n$" So I'm stumped ...
0
votes
1answer
42 views

Integral of $sin|x|$

$$\int \sin|x| dx$$ We have two cases: x less than zero, or x equals or higher than zero. $$\int_{-\infty}^0\sin(-x) dx + \int_0^{\infty}\sin x dx$$ Left side of this sum is equals to right side, so ...
0
votes
2answers
39 views

Integrating $g: ℝ^2\to ℝ$ - Order of Integration

The problem: My work: I found the two integrals to be equal to each other, which is clearly not the desired result. Any suggestions/pointers? Thanks!
0
votes
3answers
25 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
0
votes
2answers
43 views

How do you find the derivative of the $\int_{1-x}^{1+2x} e^{t^2} dt$?

$$\int_{1-x}^{1+2x} e^{t^2} dt$$ I don't fully understand the steps taken to answer this question. If someone could please tell me the steps for this kind of a situation I would really appreciate it. ...
2
votes
1answer
84 views

Simple integral $\displaystyle\int \frac{e^x}{x^2-a^2}\ dx$

Is this integral solvable? $$\int \frac{e^x}{x^2-a^2}dx,\quad a>0.$$
0
votes
2answers
43 views

Methods to do Integral

I know this integral can be done using complex analysis. Are there some slick solutions using standard calc methods? $\displaystyle\int_{-\infty}^{\infty}\displaystyle\frac{1}{(x^2+1)(x^2+9)}dx$
0
votes
0answers
6 views

Integration of characteristic function with varying boundaries

I'm a bit puzzled about integrals with indicator/characteristic functions in them. How do I start computing the following integrals? $$ A\int_{-\infty}^{\infty}f(x)\chi_{[-a+x,a+x]}dx $$ and $$ ...
0
votes
0answers
15 views

Question on the Prime Number Theorem (the Tchebychev Function)

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
2
votes
0answers
37 views

Is there a generalization of integration by parts?

here is what i concerned: there are $u(x)$ and $v(x)$ in the original integration by part formula, what if the integral involve with one more function $w(x)$. Second of all, i know that there are ...
0
votes
1answer
17 views

Changing the domain of integral

I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg: Why ...
1
vote
1answer
61 views

Derive The Midpoint Rule: $\int_{x0}^{x1}f(x)dx=hf(x_0+\frac{h}{2})+\frac{h^3}{24}f^{2}(\mu)$

The Given Question is: ================================================================== Expand the function $f(x)$ in a $1^{st}$ degree Taylor series about $x_0 + \frac{h}{2}$ with ...
0
votes
4answers
57 views

How can I prove the integral?

Prove that $$ \int\frac{dx}{x(\log_e x)^{7/8}} = 8(\log_e x)^{1/8} $$ I am totally lost on this subject. Any help how to prove this is appreciated!
6
votes
3answers
630 views

How do I solve this definite integral?

$$\int_0^{2\pi} \frac{dx}{\sin^{4}x + \cos^{4}x}$$ I have already solved the indefinite integral by transforming $\sin^{4}x + \cos^{4}x$ as follows: $\sin^{4}x + \cos^{4}x = (\sin^{2}x + ...
1
vote
2answers
32 views

Integrating a Partial Derivative

Would I be right to think that $$\int dx \,\,\,\frac{\partial}{\partial x} f(x,y)=f(x,y)$$ Or are there pathological cases?
-1
votes
0answers
18 views

How to integrate this function and decide lambda

I want to decide lambda for I also want to integrate the same function. Please help!
0
votes
1answer
38 views

Double Integral Proof

Let function $f(x, y)$ be defined by $$f(x, y) =\begin{cases} 1,\text{ if }x = y,\\ 0,\text{ otherwise}.\end{cases}$$ Using the definition of the double integral show that the following integral exists ...
0
votes
1answer
50 views

Integral of $e^x(1-x)^n$

$$I_n=\int_0^1 e^{x}(1-x)^n \;\text{d}x \;\;\; \text{ for } n\in \mathbb{N}$$ Evaluate $I_0$. Can someone help me out how to do this. Am I right in saying that the integral of $e^x = e^x$? I ...
3
votes
1answer
60 views

Does this integral have any closed form? $\displaystyle\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$

Does this integral have any closed form? $$\int\frac{1}{x+\sin(x+1)}\mathop{\mathrm dx}$$ I think the substitution $x=(u-1)+2\pi$ will do it, no?
0
votes
2answers
74 views

Is it possible to convert $\sigma = \int_0^\infty e^{-x^2} dx$ to an integral problem over $(0,1)$? [on hold]

Is it possible obtain a transformation to convert $\theta=\displaystyle\int_0^\infty e^{-x^2}\, dx$ to an integral problem over $(0,1)$?
2
votes
2answers
50 views

Evaluating integral limit in two ways gives different limits

Problem Show that $$\lim\limits_{h \rightarrow 0^{+}} \int_{-1}^{1} \frac{h}{h^{2}+x^{2}} \, dx = \pi.$$ I can do this by evaluating the integral directly and showing that it is equal to ...
0
votes
1answer
37 views

Strange Dirac delta distribution contradiction

Consider the following integrals in variables $x,y$ over the whole $\mathbb{R}$, where $a,b\in\mathbb{R}/0$ are constants: $$\int dx \int dy ~\delta(x-a)\delta(y-b\,x)=\int dy ~\delta(y-b\,a)=1$$ In ...
-3
votes
0answers
38 views

$\int_{a}^{b}{x^nf(x)dx}=0$ for all $n$ [on hold]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Prove that if $\int_{a}^{b}{x^nf(x)dx}=0$ $\forall n\in \mathbb N$ then $f(x)=0$ for all $x\in [a,b]$
1
vote
0answers
15 views

Question concerning the domain of polar coordinate.

So in the problems I encountered, I find it confusing about the domain of $\theta$. Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T ...
1
vote
1answer
14 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
0
votes
0answers
23 views

$\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ where $M$ is the sup [duplicate]

Let $f:[a,b]\to \mathbb R$ be a continuous function. Let $\displaystyle M=\sup_{x\in [a,b]}{f(x)}$. Prove that: $\displaystyle \lim_{n\to \infty}{(\int_{a}^{b}{(f(t))^ndt})^{\frac{1}{n}}}=M$ I ...
0
votes
0answers
15 views

Mean value of a function over the n-sphere superficie.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
1
vote
1answer
37 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
0
votes
2answers
26 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
3
votes
4answers
136 views

How to show $\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$? [duplicate]

How to show this equation below is true? $$\int_{0}^{\infty}e^{-x}\ln^{2}x\:\mathrm{d}x=\gamma ^{2}+\frac{\pi ^{2}}{6}$$ Where $\gamma$ is the Euler-Mascheroni constant....
0
votes
1answer
15 views

Interpolation of Polynomial using Lagrange

$f(x) = x^3 + 2x^2 + x + 1$. Find a polynomial of degree $4$ that interpolates the values of $f$ at $x = -2, -1, 0, 1, 2$. I was trying to use the Langrange algorithm, but I think i'm doing it ...
0
votes
1answer
40 views

Integration Question Help

How to integrate the following function: $$\int_{a}^{b} \frac{\sqrt{1-y^2}}{1+y^2} dy$$ -----------------------
1
vote
2answers
45 views

Proving that the line integral $\int_{\gamma_{2}} e^{ix^2}\:\mathrm{d}x$ tends to zero

Let $f(z) = e^{iz^2}$ and $\gamma_2 = \{ z : z = Re^{i\theta}, 0 \leq \theta \leq \frac{\pi}{4} \} $. All the sources I have found online, says that the line integral $$ \left| \int_{\gamma_2} ...
0
votes
2answers
19 views

Interpolation of Polynomial

Let $f(x) = x^3 + 2x^2 + x + 1$. Find the polynomial of degree $2$ that interpolates the values of $f$ at $x = -1,0,1$. I was able to do the an initial part of this problem (not written), but I ...
1
vote
1answer
29 views

Trapezoid Rule - Number of Points

How many points should we use in the trapezoid rule in computing an approximate value of $\int_{0}^{1} e^{x^2} dx$ if the answer is to be within $10^{-6}$ of the correct value? I'm looking at the ...
2
votes
1answer
30 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
0
votes
1answer
12 views

Polynomial Interpolation - Bound on Error

Let the function $f(x) = \ln(x)$ be approximated by an interpoation polynomial of degree of 9 with 10 nodes uniformly distributed in the interval $[1,2]$. What bound can be placed on the error? I've ...
0
votes
1answer
26 views

Natural Cublic Spline Confusion

Find the natural cubic spline which interpolates the data points $(1,0),\; (2,1),\; (3,0), \; (4,1), \; (5,0) $. I know how to check if a piecewise function is a natural cubic spline, but I don't ...
0
votes
0answers
15 views

Find the volume of this region.

Find the volume of the region bounded by $ (x^2+y^2+z^2)^2=x$. I do not know how to deal with this question. Please help.