Questions about the evaluation of specific definite integrals.

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54 views

How should I try to evaluate this integral?

Suppose that we are given the following integral: $$\int_0^\pi \sqrt{1+4\sin^2\frac x2 - 4\sin\frac x2}\;dx.$$ (Original screenshot) And the answer is one of these :- $4\sqrt3-4-\frac\pi3$ ...
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0answers
72 views

Beautiful Closed form $\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$

Hi I am trying to calculate $$ I:=\int_0^1 dx \frac{\ln x \ln^2(1-x)\ln(1+x)}{x}$$ Note, the closed form is beautiful and is given by $$ I=−\frac{3}{8}\zeta_2\zeta_3 -\frac{2}{3}\zeta_2\ln^3 2 ...
2
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2answers
51 views

Find the length of $y=x^2$ from $x=0$ to $x=1$

$$L=\int_0^1 \sqrt{1+4x^2} dx = \frac{1}{4}\left(\theta+2\sqrt{5}\right)$$ It tells you to use $2x=\sinh\theta$. I've tried the best I could, but I can never get the right side of the equation. Can ...
2
votes
2answers
44 views

Problem calculating line integral

I have $\gamma=[0,1]\to\mathbb{R}^3$ defined by $\gamma(t)=(\cos(2\pi t), \sin (2\pi t), t^2-t)\;\forall t\in[0,1]$ and I'm asked to calculate ...
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1answer
41 views

A problem about center of mass

Suppose $f(x)$ is positive, increasing and Riemann-integrable on the interval $[a,b]$. Let$$\bar{x}=\frac{\int_{a}^{b}{xf(x)\text{d}x}}{\int_{a}^{b}{f(x)\text{d}x}}.$$Prove ...
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0answers
62 views

How to solve this integral ($\int _{\frac{\pi }{6}}^{\frac{\pi }{4}}\sqrt{1-\tan ^2\left(x\right)}dx$)

$$\int _{\pi/6}^{\pi/4}\sqrt{1-\tan ^2\left(x\right)}dx$$ Hey, can you help me to solve this integral please? Thanks.
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38 views

Computing the values of $ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{2n} $

Define the following integral as $$ W_\alpha(n):=\int_0^\pi x^{\alpha-1}\sin(x)^{n}\,,\quad V_\alpha(n):=\int_0^\pi x^{\alpha-1}\cos(x)^{n} $$ where $n \in\mathbb{N}$. Now in the base case ...
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28 views

Fundamental Theorem of Calculus and inverse..

If $F(x)$ is defined as $$F(x)= \int_{a}^{x} f(t) dt$$ calculate $(F^{-1})'(y)$ in terms of $f$. I have been working on this for a while now, does the aanswer to this incorporate the Inverse ...
2
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1answer
51 views

Integration by expansion

Consider the integral \begin{equation} I(x)= \frac{1}{\pi} \int^{\pi}_{0} \sin(x\sin t) \,dt \end{equation} show that \begin{equation} I(x)= \frac{2x}{\pi} +O(x^{3}) \end{equation} as ...
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1answer
31 views

a question about integral? I have no idea about that!

If f(x) and g(x) are integrable in [a,b], can we say that f(x)g(x) is still integrable in [a,b]? I am referring to Riemann integration!
2
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1answer
30 views

Re-interpreting double integral as a Type II Region $\mathrm{d}y\,\mathrm{d}x$ vs $\mathrm{d}x\,\mathrm{d}y$

I have the following Double Integral:$\iint_Dx\cos y\space\mathrm{d}A$ where $a$ is bounded by $x=1,y=0,y=x^2$. Interpreting this region as a Type one region, it is easy to conclude $R=\{(x,y)\mid ...
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1answer
36 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
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0answers
26 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 ...
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1answer
58 views
+50

The Fourier transform of a power of the absolute value function (and a related integral)

What (Fourier-analytic?) methods would I use to compute the following two integrals? $\displaystyle\int_{\mathbb{R}} e^{2 \pi i t} |t|^a dt \:\:\:\:\:\:\: \:\:\:\:\:\:\: \text{ and } ...
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37 views

Log Cosine Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a ...
2
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2answers
35 views

Definite integral (mixture of functions)

I have problem with this integral and I generally don't know how to approach it: $$\int_{-2}^2 (x^4+4x+\cos(x))\cdot \arctan\left(\frac{x}{2}\right)dx$$ I know that I probably have to make some ...
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1answer
128 views

How to prove $\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}$?

How do you prove that $$\int^1_0\int^1_0\frac{\log(x-x^2)-\log(y-y^2)}{(x-x^2)-(y-y^2)}dxdy=7\sum_{i=1}^\infty i^{-3}\;\;\;\left(=7 \zeta(3)\right)~?$$ p.s. Mathematica gives a pretty good ...
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+400

$\int_0^1 [ \frac{1}{x(x-1)} (2\mathrm{Li}_2(\frac{1-\sqrt{1-x}}{2})-\log(\frac{1+\sqrt{1-x}}{2})^2 ) -\frac{\zeta(2)-2\log^2 2}{x-1} ]dx$

Hi I am trying to evaluate $$ I:=\int \limits_{0}^{1} \left[ \frac{1}{x(x-1)} \bigg(2\mathrm{Li}_2\bigg(\frac{1-\sqrt{1-x}}{2}\bigg)-\log\bigg(\frac{1+\sqrt{1-x}}{2}\bigg)^2 \bigg) ...
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3answers
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For solid volumes, why does the Integral behave as a summation?

When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation) Now, when you take the volume of a 3D object, you sum the ...
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3answers
68 views

$\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$

Evaluate $\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$ Sidenote:Via mathalpha I know that answer is $-\pi/4$ but do not know how to derive that.
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1answer
47 views

Expansion of Integration

Consider the integral \begin{equation} I(x)=\int^{2}_{0} (1+t) \exp\left(x\cos\left(\frac{\pi(t-1)}{2}\right)\right) dt \end{equation} show that \begin{equation} I(x)= 4+ \frac{8}{\pi}x +O(x^{2}) ...
2
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2answers
58 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
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1answer
33 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
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2answers
35 views

Short argument for asymptotic value of parameter integral

I want to find the main term of the asymptotic expansion for $x\to 0^+$ for $$f(x)=\int_0^{\pi/2} \dfrac{\cos t}{t+x}dt.$$ Now, clearly, the problem is at $t=0$ and the cosine is almost $1$ ...
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3answers
103 views

Showing that $\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$

I was reading an article in which it was stated that, with a change of variable, one could show that: $$\int_{0}^{\infty} \frac{dx}{1 + x^2} = 2 \int_0^1 \frac{dx}{1 + x^2}$$ I tried with $t = 1 + ...
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1answer
34 views

Help me integrate this function using Simpson's rule

I have a question: compute $$\int_0^1 \frac{\sin(x)}{x}\,dx$$ for $n=10$ divisions. I got the value $0.9127$ but I think its a bit too high.
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15 views

Argument Principle solving the quesion

1/2pi i ∫ c(0,12) f'(z)/f(z) dz where f(z) = (z-5+12i)^4(z-7)^6 / 12(z-8)^6(z-i+6)^7 dz. How do I go about doing this kind of question. I only know its argument principle. I googled it but still cant ...
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1answer
50 views

Integral $\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt$

I am looking for a closed form expression for the logarithmic trigonometric integral $$ I_{n,p}=\int_0^{\pi/2} \log^n (\sin t)\log^p (\cos t) dt \quad (n\geq 0, p\geq 0). $$ Closed form expression ...
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2answers
96 views

Integrate $I=\int_0^1\frac{\ln x}{x^n-1}dx$

Hi I am trying to obtain a closed form for$$ I_n=\int_0^1\frac{\ln x}{x^n-1}dx, \quad n\geq 1. $$ This integral is quite nice and generates many other known closed form results such as $$ ...
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30 views

Integral $I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx$

Hi I'm trying to show that $$ I=\int_0^1 \frac{\arctan\big(\sqrt{x^2 + 2}\big)}{\sqrt{x^2 + 2}(x^2 + 1)}dx=\frac{5\pi^2}{96}. $$ We can try the substitution $u=(x^2+2)^{1/2}, du=x(2+x^2)^{-1/2}dx$ ...
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1answer
136 views

Calculate $\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$

I am trying to calculate: $$\int_0^1 \frac{\ln(1-x+x^2)}{x-x^2}dx$$ I am not looking for an answer but simply a nudge in the right direction. A stradegy, just something that would get me started. ...
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1answer
48 views

How to integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$?

How do I integrate $\int \frac{dy}{\sqrt{4y+\frac{1}{4y^2}+2C_1}}$, where $C_1$ is an arbitrary constant? Is this integral really complex (hard to integrate)? EDIT: This comes from DE: $dy/dx = ...
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3answers
25 views

Integral $\int_{0}^{\pi}\sqrt{1+4\sin^{2}(x/2)-4\sin(x/2)}\mathrm{d}x$

Here's how I solved it. \begin{eqnarray*} & & \int_{0}^{\pi}\sqrt{1+\left(4\sin^{2}\left(\frac{x}{2}\right)\right)-\left(4\sin\left(\frac{x}{2}\right)\right)}\mathrm{d}x\\ & = & ...
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1answer
26 views

How do I integrate this in terms of error function

How do I evaluate $$\dfrac{1}{\sqrt{4\pi t}}\int_0^{\infty}ye^{-\frac{(\xi-y)^2}{4t}}dy$$ in terms of $\text{erf}(x)$ ? I tried integration by parts but the integral seems to get complicated. I think ...
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1answer
40 views

integration $\int_a^b$$f$$(x)$$dx$ when $f(a)=f(b)$

How to integrate $\int_a^b$$f$$(x)$$dx$ when $f(a)=f(b)$ Can something like $\int_0^kf(x)dx=2\int_0^{k\over2}f(x)dx$ for $f(x)=f(k-x)$ be somehow used?
3
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3answers
228 views

William Lowell Putnam Integral Problem

Prove That $$ \frac{22}{7}-\pi= \int_0^1 \frac{x^4\,\left(1-x\right)^4}{1+x^2}$$
5
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1answer
65 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 ...
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2answers
54 views

Definite Integral $\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$

$$\int_{-\infty}^{\infty}\frac{x\sin x}{(x^2+a^2)(x^2+b^2)}\,\mathrm{d}x$$ This is easy to evaluate with complex analysis but is there an elementary way (substitution, partial fractions, integration ...
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1answer
47 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!
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81 views
+400

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 $$ I=\frac{1}{\pi}\int_0^{\pi/3}\log\left( \frac{1+2\cos\theta}{2}+\sqrt{\bigg( \frac{1+2\cos\theta}{2} \bigg)^2-1} \right)d\theta. $$ ...
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3answers
30 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
34 views

Showing $f(z_0) = \frac{1}{2 \pi} \int_0^{2 \pi} f(z_0+Re^{i\theta}) \ d\theta$

Suppose that $f$ is analytic on and inside the circle $|z-z_0|=R$. Show carefully that $f(z_0)$ is equal to the average of $f$ on ${|z-z_0|=R}$, i.e. show that $$f(z_0)={1 \over {2 \pi}} \int_0^{2 ...
1
vote
1answer
35 views

Bessel's integral, how to actually evaluate?

I am just about to study Bessel functions and I have recently seen one of its integral representations given by: $$ J_ \alpha (x) = \frac{1}{\pi} \int_0 ^ \pi \cos(\alpha \tau - x\sin\tau) d\tau - ...
7
votes
3answers
673 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 + ...
0
votes
1answer
39 views

Proof of integral equality

Let $f^{(n)}(x)$ be the $n$-th derivative of $f(x) = \cos(x)$. Prove that : $$ \int_0^{2\pi} f^{(n)}(x) \,\, dx = \int_0^{2\pi} f^{(n)}(kx) \,\, dx, $$ where $n$, $k$ are natural numbers equal or ...
0
votes
2answers
40 views

Integral of $e^x ln(e^{2x} - 4)$

Find the integral from ln4 to ln6 of $$e^x \ln(e^{2x} - 4)$$ I factored $$\ln(e^{2x} - 4)$$ to get $$\ln((e^{x} - 2)(e^{x} + 2))$$ Then I separated this to get: $$e^x\ln(e^{x} - 2) + e^x\ln(e^{x} + ...
2
votes
1answer
33 views

A bounded integral

I want to show that there exists $K\in\mathbb{R}^+$ such that $$\left|\int_{1}^x \sin(t+t^7)dt \right|<K$$ for all $x\ge 1$. Intuitively, I'm quite sure it is true, but I can't find a formal proof. ...
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1answer
36 views

Integral Involving $(\arcsin x)^3$

Find the integral of $$\int_{0}^{1}8(\arcsin x)^3 dx$$ Okay, so I have substituted $u = \arcsin x$. I multiplied the integral by $$(1-x^2)^{1/2}/(1-x^2)^{1/2}$$ And then I said that $$\cos u = ...
1
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1answer
43 views

Complex contour integral with sign function:$-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dp$

I am trying to evaluate the integral: $-i \int \limits_{-\infty}^\infty \frac{{\rm sgn}(x)^2 ~x~ e^{i x}}{1+ax^2} dx$ with sgn$(x)$ the sign function and $a$ positive real. Naively applying the ...
1
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2answers
43 views

Area under the curve $f(x) = \sin x$

Find the area under the curve $f(x) = \sin x$ on the interval $[0, \pi]$ if $\sin x \ge 0$ My handbook give this as $$\int_0^\pi \sin x \space dx = (\cos \pi) - (\cos 0) = (-1) - (-1) = 2$$ what ...