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.

20559 questions

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Is $f(x) = \frac{\sin(x)}{x}e^{-xy}$ integrable for $x\in ]0,\infty[$ and $y \geq 0$?

Is $\left]0,\infty\right[ \to \mathbb{R},\,\,\, \operatorname{f}\left(x\right) = \frac{\sin\left(x\right)}{x}\,\,{\rm e}^{-xy}\,\,\,$ Lebesgue-integrable for $\,\,\, y \geq 0\ ?$. I tried $\,\,\,\left\vert\operatorname{f}\left(x\right)\right\vert…

definite-integrals

asked Jan 10 '21 at 00:38

Mathpiano

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2 answers

Is $\int_{0}^{\infty}\frac1{\sqrt{x^{2n}+1}}dx$ finite?

I want to evaluate the integral $\int_{0}^{\infty}\frac1{\sqrt{x^{2n}+1}}dx$, where $n\in \mathbb{N},n\ge2$ in order to show the integrability of another function. However, this seems rather difficult, but I need only to know whether it has a finite…

definite-integrals

asked Jan 09 '21 at 20:43

Mathpiano

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3 answers

Evaluate $\int_{2}^{7} \frac{x}{1-\sqrt{2+x}} d x$

We have the following integral: $$ \int_{2}^{7} \frac{x}{1-\sqrt{2+x}}\, dx $$ And this is my solution, which seems to be wrong, and I am failing to see where exactly I failed at: We have $u=1-\sqrt{2+x}, x=u^2-2u-1, dx=-2\sqrt{2+x}\, du$, and we…

definite-integrals

asked Jan 06 '21 at 18:14

zareami10

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3 answers

Find the area bounded by the curve $y=\ln x$, $y=\frac xe$ and the x y axis

$\ln x$ intersects the y axis at $-\infty$ So $$A=\int_{-\infty}^{e} \ln x \,dx-\int_{-\infty}^{e} \frac xe dx$$ Which isn’t really a proper definite integral since I am having trouble applying the limits What is the correct way to do it?

definite-integrals

asked Dec 20 '20 at 11:25

Aditya

6,191

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How to do the integration?

While reviewing an article "A note on the wall-jet problem" by MERKIN and NEEDHAM, I found one integration a bit annoying. Need a little help to sort it out. We have an equation \begin{equation} \int_{0}^{\infty} \left(f(x)-\alpha\right)f'^2(x)…

definite-integrals

asked Nov 30 '20 at 14:36

Waqar

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1 answer

Using Triple Integrals to Compute Total Mass

The question I have is as follows: A rectangular swimming pool with vertical sides that measures 50ft by 150ft has a depth that increases in linearly from the short end (3 feet deep) to the deep end (15 feet deep). This is assuming that water…

definite-integrals

asked May 15 '13 at 16:17

duxrule

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1 answer

If $I_n=\int_{-\pi}^{\pi} \frac{\sin nx}{(1+\pi^x)(\sin x)}dx$, then prove the following

$I_n=I_{n+2}$ $ \sum_{m=1}^{10} I_{2m+1} =10 \pi$ $\sum _{m=1}^{10} I_2m =0$ My simplification of the given integral is $$I_n =\int_0^{\pi} \frac{\sin nx}{\sin x} dx$$ Where $n$ is an integer I believe the expression needs further…

definite-integrals

asked Nov 18 '20 at 16:26

Aditya

6,191

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how can I do the following integration?

$$ \int_0^h\!\exp\left( ity + (\alpha-1) \ln y \right)\,dy, $$ where $i = \sqrt{-1}$, $\alpha>0$, and $h>0$. Please show your work step by step. I know $$ \int\! \exp \left( mx \right)\,dx = \frac{\exp ( mx )}{m} + c, $$ where $c$ is a…

definite-integrals

asked May 12 '13 at 05:38

time

1,595

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4 answers

Does $\int_{-a}^af(x)\,\text{d}x=0$ implies $f$ is odd?

We know if a function is odd then $\int_{-a}^af(x)\,\text{d}x=0$, but I am queried if the converse is true or not. I.e., if $\int_{-a}^af(x)\,\text{d}x=0$ does it imply function $f$ is odd?

definite-integrals

asked Oct 26 '20 at 10:04

Soubhagya

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1 answer

Intuitively, why is the double integral also the area under the curve?

I just learned that the single variable definitely integral is the area under the curve via the Riemann's sum intuitive, but now I'm lost as to why the double integral also represents an area? What is going on? Like in the last example, isn't the…

definite-integrals

asked Oct 06 '20 at 13:09

Jwan622

5,704

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2 answers

An integral of an absolute value of sinusoidals

Does anyone know how to obtain an expression for the following integral: $$ \int_{0}^{2\pi}\int_{0}^{2\pi} \mathrm{e}^{\mathrm{i}\alpha\left\vert\,{\sin\left(\,{\theta}\,\right) -…

definite-integrals

asked Oct 02 '20 at 14:53

Chris

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1 answer

Basic integral proof

I have the following statement: Prove that if $f$ is continuous then $\int_a^b{f(-x)dx}= \int_{-b}^{-a}f(x)dx$ My attempt was: Starting from LHS, Let $u=-x$ therefore i have $-\int_{-a}^{-b}f(u)du=\int_{-b}^{-a}f(u)du$. But from here i don't know…

definite-integrals

asked Sep 28 '20 at 15:01