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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Show that $\int_{\sqrt{k\pi}}^{\sqrt{(k+1)\pi}} \sin (x^2) \, dx = \frac{(-1)^k}{\xi_k}$ for a suitable $\xi_k$

Let $k \in \mathbb{N}$. Show that there exists $\sqrt{k\pi} < \xi_k < \sqrt{(k+1)\pi}$ such that: $$\int_{\sqrt{k\pi}}^{\sqrt{(k+1)\pi}} \sin(x^2) \, dx = \frac{(-1)^k}{\xi_k}$$ We haven't introduced Fresnel integrals, which might be useful here...…
looki
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How to obtain the following definite integral?

I have no idea how to obtain the answer to the following problem: $$I=\int_{0}^{1}\frac{dx}{\sqrt{8-x^2-x^3}}$$ The answer has been given as $$\sin^{-1} \frac{1}{2\sqrt{2}}< I < \frac{1}{\sqrt{2}}\sin^{-1} \frac{1}{2}$$ Any help will be appreciated.…
userNoOne
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Is $\int^0_0 x^{-1} dx$ defined?

Is an object like $$\int^0_0 x^{-1} dx$$ defined? Context: when considering functions such as $$f(x)=\int^{x^2}_xt^{-1}dt$$ would it therefore be necessary to give a piecewise definition of $f$ if we wished to include $0$ in the domain? A little…
Shuri2060
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Integral with bounds

I have define integral: $$\int_0^\pi\frac{x\sin x\,dx}{1+\cos^2x}=\frac{\pi}{2}\int_0^\pi\frac{\sin x\,dx}{1+\cos^2x}$$ my question is how do we get that $\pi/2$
user5927353
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Does this one require integration by parts?

$$\int_{0}^{1} x \sqrt{1+8x^2}\,dx$$ I tried $u=x$ $\text{d}u=\text{d}x$ $\text{d}v = (1+8x^2)^\frac{1}{2}$ but I'm not sure how to get $v$ by integrating $dv$, since $u$-sub doesn't work. Barking up the wrong tree?
JackOfAll
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Is there any hope for an analytical solution of this integral?

In the simulation of a physical process, I face the problem of computing $$S=\int_{\tau_1}^{\tau_2}\int_{z_1}^{z_2}\frac{e^{-\large{\frac{a z^2+b z+c}{\tau }}}}{\tau ^{3/2}}\,d\tau\,dz$$ (I leave the problem as it is set; for sure, completing the…
Claude Leibovici
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How to change integral bounds?

In the following integral I want to change the bounds from $(0, 2)$ to $(-1, 1)$: $\displaystyle{\int_{0}^{2}(1+x)^3 dx}$ How do I change them? I know that a variable changing is needed, but don't know how to use it to change the bounds. Indeed my…
user16948
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Determine a positive integer $n\leq5$,such that $\int_{0}^{1}e^x(x-1)^ndx=16-6e$

Determine a positive integer $n\leq5$,such that $\int_{0}^{1}e^x(x-1)^ndx=16-6e$. I tried to solve it.But since $n$ is given to be $\leq$ 5,my calculations went lengthy. Applying integration by parts repeatedly,we get \begin{align} \int e^x(x-1)^n…
Brahmagupta
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Find the value of $\int_{1}^{e} \frac{\ln x}{x+1}dx$

Find the value of $$\int_{1}^{e} \frac{\ln x}{x+1}dx.$$ I don't have solution for this problem. Can you help me?
Road Human
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Computing an integral arising in potential theory

UPDATE I have a conjecture as to the solution, based on some theoretical considerations, and it holds at least for $d=3,4$: $$ \int_0^\pi \left( \frac{\sin\phi}{\sqrt{1+u^2-2u\cos\phi}}\right)^{d-2}\,d\phi = \frac{\omega_{d}}{2\omega_{d-1}}…
Eugene Shvarts
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Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

I'm looking to calculate $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$. Mathematica10 can't integrate this, but numerical integration gives an answer of 5.50843 for any $a$ I've tried, so I assume that there…
Dave
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Integration of $\int_0^{2\pi}(e^{\cos x}\cos x\sin x) \,dx$

Can anyone please help me with this integration: $$\int_0^{2\pi}(e^{\cos x}\cos x\sin x)\,dx$$ I am getting the answer as $0$ by using simply the properties of definite integral BUT the answer as told by my teacher is $2\pi$ Please help. Thanks!
NeilRoy
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Integral of $R(R^2+y^2)^{-3/2}$ with respect to $y$

$$\int_0^\infty \frac{R}{\sqrt{R^2+y^2}\left(R^2+y^2\right)}dy$$ The indefinite integral seems to be $$\frac{-R}{\sqrt{R^2+y^2}}+C$$ $R$ is a constant
Nikita
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How to calculate the integral of $\operatorname{sgn}(\sin\pi/x)$ in the interval $(0,1)$?

How can I calculate the integral of $\operatorname{sgn}(\sin\pi/x)$ in the interval $(0,1)$? I need to calculate this integral, thanks
sergio
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Equality of two integrals

Let $a\in\mathbb{R}$ and $f:[a-1,a+1]\longrightarrow\mathbb{R}$ be a differentiable function, for which $f(a-t)=f(a+t)$ for all t $\in[-1,1]$. Prove, that $$\int_{a-1}^{a+1}xf(x)\ dx=2a\int_a^{a+1}f(x) \ dx\ .$$ I know I need a clever substituion,…
Jules
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