# Tagged Questions

8 views

### Integration Question - Not sure how to approach

I have absolutely no idea how to approach this question: $$\int \frac{x^2}{(15+6x-9x^2)^{3/2}} \ \mathrm{d}x$$ I'm almost positive that it has something to do with trigonometric substitution, but ...
4 views

### Analytical Formula for Hilbert Transform of a Ricker Wavelet

I am attempting to validate some numerical code I have to compute Hilbert transforms. As I am interested in the Hilbert transforms of functions with rapid decay, I wanted to unit test my code with the ...
43 views

### Integration with substitution $u=\cos(x/2)$

Please could you help me solve this integral Find $$\int \cos x \, \sqrt{1-\cos x} \, dx.$$ Hint: use the substitution $u=\cos(x/2)$. Thanks.
39 views

### Evaluating an integral in calculus III

Please can someone explain what happened after step 3 , in this upload image of the exercise Please make it clear how the "1/2" came in step 4 , and why we are subtracting the integrals
84 views

### Value of this definite integral $\int_{0}^{\infty} \frac{ \ln(x)}{x^2+2x+4} dx$

So I came across this question on brilliant.org and didn't know how to go about it: $$\int_{0}^{\infty} \frac{ \ln(x)}{x^2+2x+4} dx$$ I tried to complete the squares in the denominator and then use ...
38 views

### $\int_0^{1/2} \cos^{-1} x \, dx$ using integration by parts.

integral lower bound is 0. upper bound is 1/2. function is cos^-1 x dx. My work: I use integration by parts. u = cos^-1 x ... du = -dx/sqrt(1-x^2) ... v = x ... dv = dx ... so integral udv = ...
20 views

### Is there any integration of a defined function that could not be express as a convergent infinite series?

Is there any integration of a defined function that could not be express as a convergent infinite series?Like if it could be getting a divergent series as an answer I wonder if the answer is a yes or ...
21 views

### why we say this function have closed form while the other doesn't?

why we say this function have closed form while the other doesn't? $\int(sin(x)) dx=-cos(x)+ C$ have a closed form While $\int sin(x)/x$ dx = Si(x)+ C does not have a closed form?
27 views

### What is list of common integral that have no closed form?

What is list of common integral that have no closed form? It's diffucult for me to google it for some reason.
44 views

### Differentiability of function defined as integral

Suppose $$F(x) := \int_0^1 f(t,x) dt$$ is well-defined for all $x \in \mathbb{R}$. I would like to show that $F(x)$ is not differentiable at $0$. Is it enough to show that $\partial_x f(t,0)$ is ...
47 views

### Integration involving $\log_2(x)$

Having a hard time going about this problem: $$\int{\frac{\ln(2)\log_2(x)}{x}}$$ I believe $\ln(2)$ would be considered a constant, so than the equation would then changed to: ...
28 views

### Center of Mass with two functions

I am having trouble trying to figure out how to go about this problem. I can do problems with single variables but I can not solve this one. I think I would need to subtract the functions from one ...
66 views

### Solving an integral using Leibnitz rule

Is it possible to solve the integral $$\int_0^{\pi/2}\frac{\ln(a + b \sin x)}{(a - b \sin x) \sin x} dx$$ using Leibnitz's rule of differentiating under the integral sign? If so, how? I've tried ...
16 views

37 views

### Volume of revolution of cardioid

The parametric equations of a cardioid are $x=\cos\theta (1-\cos\theta)$ and $y=\sin\theta (1-\cos\theta)$, $0\le\theta\le 2\pi$. Diagram here. The region enclosed by the cardioid is rotated about the ...
60 views

### how to integrate $\frac1{2-3x^2}\,dx$ [on hold]

How do we evaluate $$\int\frac{dx}{2-3x^2}$$
18 views

### Indefinite double integral

In calculus we've been introduced first with indefinite integral, than with the definite one. Than we've been introduced with the concept of double (definite) integral and multiple (definite) ...
89 views

### Integration of $1/(1+\sin x)$

I solved it using $t=\tan(\frac{x}{2})$ substitution and got $-2/(1+\tan(x/2))+C$, but in my math book solution is $\tan(x/2-\pi/4)+C$. Are those the same expressions and if they are, how do I ...
20 views

### solve velocity equation with slop effect?

$$\frac{dp_l}{dx}-\mu_l\frac{\partial^2 u}{\partial y^2}=0$$ where $\mu_l$ and $p_l$ is the liquid phase viscosity and pressure, respectively; and $u$ is the flow velocity. The boundary ...
61 views

### Fourier transform of $\operatorname{erfc}^2\left|x\right|$

Could you please help me to find the Fourier transform of $$f(x)=\operatorname{erfc}^2\left|x\right|,$$ where $\operatorname{erfc}z$ denotes the the complementary error function.
42 views

68 views

### Integral of $1/(x^2+a^2)^{3/2}$?

What should I substitute to calculate the integral of $1/(x^2+a^2)^{3/2}$? With a being constant. Or is there a better way than substituting for this? I tried $u=x^2+a^2$ but then I'm left with a ...
48 views

### Prove that : $\lvert s_n - \frac \pi 4\rvert \le \frac 1 {2n+1}$, where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1}$

Prove (Leibniz' series): $|s_n - \frac \pi 4| \le \frac 1 {2n+1}, \forall n \in \mathbb N$ where $s_n = \sum^{n-1}_{j=0} \frac {(-1)^j} {2j+1} = 1 - \frac 1 3 + \frac 1 5$ ... To prove the result ...
138 views

### Finding g(1) where g(x) is the antiderivative of $f(x)= \sqrt[3]{x^2+4 x}$ and g(5)=7 [closed]

Let $f(x) = \sqrt[3]{x^2+4 x}$ and let $g(x)$ be an antiderivative of $f(x)$. Then if $g(5)=7$, find $g(1)$. my apologies for lack of details or missing context of the question. That is exactly how ...
20 views

### Bound for the integral over one region, given integrals over nearby regions

We have the following: $f:\mathbb{R}^2\rightarrow [0,1]$, and constants $b,c, \varepsilon_{x_k},\varepsilon_{y_k}$ such that: $$\tag{1}0<c<b$$ \tag{2} ...
107 views

### Find the integral $\int \frac{1}{x^2 \cdot \tan(x)} \ dx$

This problem seems pretty tricky. I need to find the integral of $$\int \dfrac{1}{x^2 \cdot \tan(x)} \ dx$$ Any help would be greatly appreciated!
158 views

### Integral $\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx$

I have a trouble with this integral $$I=\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx.$$ Could you suggest how to evaluate it?
Given this integral $\int_0^1\int_0^{1-(y-1)^2}\int_0^{2-x}f(x,y,z)dzdxdy$ , how can I interchange the variables and express as integrals of the other five forms like dxdydz,dxdzdy...?? So what I ...