# 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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### Find all continuous functions $f:[0,1]\rightarrow \mathbb{R}$ that satisfy: $\int_0^1 f(x)dx=1/3 + \int_0^1 f^2(x^2)dx$

(Note that $f^2(x)=f(x)\cdot f(x)$ and not composition.) Since both integrals are defined, derivation is out of the question. I tried integrating the second integral by parts but reached something ...
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### Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$?

Why are functions that are continuous over $[a,b]$ integrable over $[a,b]$? Why is it that to be Riemann-integrable the infimum of the upper sums and the supremum of the lower sums have to be equal? ...
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I am trying to find a parametric equation for the intersection line of the surface of two orthogonal cylinders, $\vec{P}$ is a point that belongs to this intersection: $$\vec{P(t)} = \begin{bmatrix} ... 0answers 14 views ### Is there any equality for the integral of the product of normal derivative? I am trying to get the proof of \int\int_DD_uf(x) D_ug(x) dx. For example in Green Theorem, in integral we use the product of  \nabla, when it comes to normal derivative, how can I organize the ... 0answers 27 views ### Calculate the curvilinear integral I need to calculate the curve integral. This should be the curve integral of I rank, which can be calculated with the formula :$$\int_{C}f(x,y)ds=\int_{a}^{b}f(g(t),h(t)) \sqrt{(\frac{dx}{dt})^2+\...
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I know there is a Proposition: for $f(x)$ is bounded on $[a,b]$,then $f(x)$ is integrable if and only if given $\epsilon>0$ ,there exists a partition such that $U(f,P)-L(f,P)<\epsilon$ But my ...
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### Evaluation of Irrational Integral

Evaluation of $$\int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx$$ $\bf{My\; Try::}$ Let $$I = \int\frac{x^4}{(1-x^4)^{\frac{3}{2}}}dx = -\frac{1}{4}\int x\cdot \frac{-4x^3}{(1-x^{4})^{\frac{3}{2}}}dx$$ ...
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### Drawing large rectangle under concave curve

Let $f$ be a continuous concave function on $[0,1]$ with $f(1)=0$ and $f(0)=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k\cdot \int_0^1f(x)dx$, ...
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### Evaluate integral over path using parametrisation

Evaluate the integral of $(F.dr)$ over the path (0,0,0) -> (1,1,1) where $F=e^{-x}i +e^{-y}j + e^{-z}k$ using parametrisation [x=t, y=t, z=t] I know from simpler questions in class that you must find ...
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### Average value of $f(x)=\int_x^1 \cos(t^2) dt$ on the interval $[0,1]$.

Find the average value of the function $$f(x)=\int_x^1 \cos(t^2) dt$$ on the interval $[0,1]$.
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### Compute antiderivative of $x \frac{3}{4} \sqrt{ -x}$

I need to compute the area under $x \times\frac{3}{4} \times \sqrt{|x|}$ from x=-1 to x=1. Therefore I need to compute the antiderivative of $x \times \frac{3}{4} \times \sqrt{x}$ for $0 \le x \le 1$,...
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### Drawing large rectangle under curve

Let $f$ be a continuous nonincreasing function on $[0,1]$ with $f(1)=0$ and $\int_0^1 f(x)dx=1$. Does there exist a constant $k$ for which we can always draw a rectangle with area at least $k$, with ...
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### Changing to Polar Coordinates for Area between $2$ Tangent Circles

I would like to calculate a double integral over: $\{(x,y) | 4x \leq x^2+y^2\leq 5x\}$. I am trying to change to polar coordinates. So the theta would go from $0$ to $2\pi$. But I am not sure what ...
### Closed form for $\int_0^1 d u \, \frac{1}{u + \lambda} \ln \left(\frac{1 + u}{1 - u} \right)$
The parameter $\lambda$ is complex and it's not on the real axis. There are some similar cases: Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$ Evaluate $\int_0^1 \frac{\ln(1+bx)}{1+x} dx$ ...