# 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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### Jacobian of the Transformation Problem, Multivariable Calculus

I have the following Jacobian problem: I'm having trouble working through it because the double integral in terms of u and v is throwing me off. Could someone walk me through it? Thanks!
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### Evaulate $\int_0^1 x\sqrt{\frac{1-x^2}{1+x^2}}dx$

I have been challanged by my teacher to solve this integral, However he gave me no hints, and I have no idea how to start $$\int_0^1 x\sqrt{\frac{1-x^2}{1+x^2}}dx$$ I noticed that putting $x=1-x$ ...
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### When Is This Technique For Dealing With Integral Singularities Valid?

I was reading the wonderful paper "Some Series of the Zeta and Related Functions" by V. Adamchik and H.M. Srivastava last night and came across an interesting technique for dealing with singularities ...
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### Evaluate $\int e^{2\theta} \sin (3\theta)\ d\theta$ [duplicate]

Evaluate $$\int e^{2\theta} \sin (3\theta)\ d\theta .$$ I am little stuck as to what I can do after this point. Please tell me if my method overall is flawed:
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### Matrix Integral Problem, what should I do

$A$ and $B$ are square matrices $$\exp(t(A+B))=\exp(tA)+\int_0^t \exp((t-s)A)B\exp(s(A+B))\,\mathrm ds$$ I found it from problem sets. It seems to define a new kind of matrix operation. I can check ...
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### Particle's displacement after 4 seconds of motion

Task The velocity, v(t) $ms^{-1}$, of an object travelling along a straight line, at time $t$ seconds is given by: $$v(t)=10e^{-\frac{1}{2}t}sin(\frac{\pi}{2}t)$$ What is the particle's ...
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### How to find $\int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x$

Is there a closed-form formula for the following integral $$\int_0^{2\pi}\log(\alpha+\beta\cos(x))\mathrm{d}x$$ where $\alpha$ and $\beta$ are constants which assure that $\alpha+\beta\cos(x)>0$...
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### Integrate $\int\frac{x+1}{(x^2+7x-3)^3}dx$

How can i solve something like that? $$\int\frac{x+1}{(x^2+7x-3)^3}dx$$ How should I start? Should I try rewrite it in partial fractions?
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### Understanding how to compute the polygonal image of this Schwarz-Christoffel mapping?

The problem statement reads: This function $\large (−z)^\frac{2}{3} (resp., (1−z)^\frac{2}{3})$ is determined as to be real and positive when $z=x<0$ (resp. when $z=x<1$) and analytic in the ...
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### If $f$ is Riemann integrable and $g$ is continuous, what is a condition on $g$ such that $g \circ f$ has the same discontinuity set as $f$?

I know that if $f$ is Riemann integrable and $g$ is continuous, then the discontinuity set of $g \circ f$ is contained in the discontinuity set of $f$. How would I go about finding a sufficient ...
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### Convergence for a improper integral $\int^b_a fg$

Let $f$ be continuous on [a,b) such that $\int^b_a f$ converges. If $g'$ is locally integrable and has a constant sign on [a,b), prove that $\int^b_a fg$ converges. Edit: We can assume that the limit ...
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### integral constant next to theta

I am a bit stuck on this practice question for an upcoming integration exam (calculus 1). Here is the integral: $$\int_ {0}^{\frac{\pi}{2}}{2\cos(3\theta)d\theta}$$ What confuses me is the 3 next ...
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### Calculating the derivative with limited info.

$$G(x) := \int_x^{x^2} f(t) \ dt$$ Calculate G'(x). I've made some progress by integrating by parts with f(t) = 1(f(t)) but I'm stuck now and don't know where to go.
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### Definite Integral involving reciprocals of logs

Integrate $$\int_2^{4e} \frac{1}{x \ln(x+1)}\,dx$$ I have tried partial fractions, u substitution and parts but i cant get the final answer out. my main problem is dealing with the $x$ and $x+1$ ...
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### Evaluate the following integral $\int_{-1/2}^{1/2}\big(\frac{\sin(n\pi f)}{\sin(\pi f)}\big)^4 df$

There are similar questions out there, but I was hoping someone could show how to would evaluate the following integral $$\int_{-1/2}^{1/2}\bigg(\frac{\sin(n\pi f)}{\sin(\pi f)}\bigg)^4 df$$ I've ...
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### Length of the curve $\;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \;\; 0 \le t \le \frac{\pi }{6}\;\;$

The length $\;L\;$ of the curve C given by $\displaystyle \;\;x=3\cos\!\left(6t\right), \;\;y=18t+3\sin\!\left(6t\right), \, \;\; \displaystyle \;\; 0 \le t \le \frac{\pi }{6}\;\;$ is found by ...