# Tagged Questions

Concerns all aspects of integration, including the integral definition and computational methods. For questions solely about the properties of integrals, use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that typically describe(s) the types of ...

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### Integration with variable in numerator

$$\int\frac{x^5}{\sqrt{25-x^2}}dx$$ I tried to do it with substitution but couldn't get ride of $x^5$ in the numerator.
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### integral problem - what is the quickest solution?

I am solving the below integral. $$\int_{0}^{1} (e^{\frac{-x}{a}}-a(1-e^{-\frac{1}{a}}))^2 dx$$ I can decompose the integrand to the simple elements doing all the algebra and then split and ...
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### Beginner Integration (Substitution)

I am pretty new to calculus and would like a nudge in the right direction in order to complete this question properly (Maybe also correct any misrepresentations I have about integration) So the ...
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### How to integrate $\int (\tan x)^{1/ 6} \,\text{d}x$? [closed]

How do I compute the following integral $$I=\int (\tan x)^{1/ 6} \,\text{d}x$$
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### Evaluate$\int_0^{\infty}x^4e^{-3x}\; dx$ using the change of variable: $t=3x$

I have been given the hint that I need to use integration by parts more than once in order to get the answer. However, I can't seem to get a reasonable result.
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### Evaluating the integral $\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx$.

How can I evaluate the following integral $$\int\frac{x^{1/6}-1}{x^{2/3}-x^{1/2}}dx.$$
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### How does the integral $\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$ converge?

I tried using the fact that $\ln(f(x)) < f(x)$ but that doesn't seem to work. It's an improper integral. $$\int_0^{\infty} \ln\left( 1+\frac{1}{x^2} \right) \,\text{d}x$$
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### How to integrate exponential and power function?

I am trying to solve the following integral $$\int_{0}^{\infty}e^{-(ax+bx^c)}\,dx ; ~~~a,b,c>0.$$ I tried using partial functions but that didn't lead to anything. Any suggestion?
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### Calculate $\lim\limits_{R\rightarrow\infty}\int_0^\pi \cos(R\cos t)dt$ w.o. Bessel function

Im calculating a complex path integral to calculate $\int_0^\infty \frac{\sin x}{x}dx$. I was able to evaluate everything except the arc $\int_0^\pi i~\exp(iR~e^{it})dt$ where $R$ is the radius. I ...
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### Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
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### Why integration constant is real?

OK, we are all taught at school that the undefined integral of a function $f(x)$ is $$\int f(x)\;\text{d}x = F(x) + k$$ where $F'(x) = f(x)$ and $k \in \mathbb R$. But, why $k$ must be real? I know ...
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### Evaluating the integral of a $\cos(\theta)$ within the exponential wrt $\theta$

I want to evaluate the following integral $\int^{2\pi}_0 \, d\theta e^{- i k (x - x')\cos{\theta}}$, where all of the variables are real and $i$ is the imaginary unit. The difficulty is the cosine ...
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### Integral Problem: $\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2}$

I am having trouble with this definite integral problem $$\int_{0}^{1} \sqrt{e^{2x}+e^{-2x}+2} \, dx$$ I know that the solution is $$e - \dfrac{1}{e}$$ I checked the step by step solution from ...
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### Related integral problem to the Gaussian integral

So according to Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \dfrac{\sqrt \pi}{2}$, $$\int_0^\infty e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$$ I want to solve for this. $$\int_0^\infty e^{-x^2}\ln(x)dx$$ ...
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### Product of Hilbert bases of $L^2(\mathbb{R}^p)$ and $L^2(\mathbb{R}^q)$ is a Hilbert basis for $L^2(\mathbb{R}^{p+q})$

Let $(\alpha_n)_n$ be a Hilbert basis of $L^2(\mathbb{R}^p)$ and let $(\beta_k)_k$ be a Hilbert basis for $L^2(\mathbb{R}^q)$. I need to show that $(\alpha_n \beta_k)_{(n,k) \in \mathbb{Z}}$ is also a ...
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### Integration of $\int_0^{+\infty} \frac{\ln(1/x)}{1-x^2}\,dx$

I am trying to integrate the below problem, but not sure if its integrable. $$\int_0^\infty \frac{\ln(1/x)}{1-x^2}\,dx$$ Had the denominator been $1+x^2$, I would have used tan(x) substitution. ...
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### evaluate a Fraunhofer diffraction integral

I need to evaluate the following integral: $$\int_{-\infty}^\infty \text{rect}(\frac{x'}{2w}) \exp\left (\frac{im}{2}\sin \left(k_Gx' \right) \right )\exp\left (-\frac{ik}{z}xx' \right )dx'$$ where ...
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### Seperation of variables algortithm.

I am doing a question for a past paper that asks me to solve using separation of variables: $y_{tt}$=$y_{xx}$-$y_x$+4y with Dirichlet zero conditions: y(0,t)=y(1,t)=0 and the initial data: y(x,0)=f(...
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### Derivative of an Integral with Two Functions as Bounds

I've been asked to find the derivative of $$g(x)= \int_{\cos x}^{x^4}\sqrt{2-u} du$$ using the Fundamental Theorem of Calculus part 1, and I know I should be substituting and setting a variable to one ...
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### Is this Complex Integration correct?

I want to integrate $\displaystyle \int_{-\infty}^\infty dx \, e^{iax}\frac{1-e^{-bx^2}}{x^2}$ for a>0. I am going to try and do this using the method of contour integration. I will choose a ...
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### Change of variables into the unit ball

Question: Express the volume of the solid bounded in $\mathbb{R}^3$ bounded below by the surface $z = x^2 + 2y^2$, and above by the plane $z = 2x + 6y + 1$, as the integral of a suitable function over ...
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### Seem to be encountering many interesting results from integrals in the form: $\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$

I have found interesting results from integrals in the form: $$I=\int_0^\infty \frac{f(a,x)}{e^{2\pi x}-1}\text{d}x$$ A few examples of interesting functions here are: f(a,x)=\sin(ax)\implies I=\...
### $\int_0^{a} x^\frac{1}{n}dx$ without antiderivative for $n>0$
My exercise is to find $\int_0^{a} x^\frac{1}{n}dx$ without antiderivatives for $n>0$. The first thing I did is plot of some of the $x^\frac{1}{n}$ for the first twenty n. This is what I got. ...
### How to test whether $e^{3x^{2}} + \frac {1}{1+3x^2} - 2\cos(x^2)$ is $o(x^3)$?
From what I learned, $\lim_{x \rightarrow 0} \frac {f(x)}{x^3} = 0$ tells $f(x) = o(x^3)$ In this case, I have tried to compute $\lim_{x \rightarrow 0} \frac {e^{3x^{2}}}{x^3}$, but the limit seems ...