Mathematics Stack Exchange
Mathematics Stack Exchange

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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Double integrate the following function

$$ \int_{0}^{a} \int_{0}^{\sqrt{a^2-y^2}} xy(x^2+y^2)^\frac{3}{2}dxdy $$ any way to evaluate the integrand tried this with multiple substitution unable to evaluate
user895466
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How to apply definite integral on $\sqrt{1+\sin x}$

I want to solve the following integral: $$\int_0^\pi \sqrt{1+\sin x} \, dx$$ I want to do this with the substitution: $$u=\sin x$$ so that: $$du=\cos x\,dx=\sqrt{1-\sin^2x}\,dx = \sqrt{1-u^2} \, dx$$ $$dx = \frac{du}{1-u^2}$$ $$u(0) = 0, \quad…
Paul
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Problem where you must find a definite integral from information given about another definite integral

I have though about u-subbing or just looking for similarities between the two integrals but those approaches seem to be getting me nowhere. I don't think I've seen a problem like this before, I am stumped. Help?
Flinn Bella
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How to integrate $\iint_D 4(x^2 + y^2) \, dxdy$ on $D = \{ |z-1| \leq 1\}$.

How to integrate $\iint_D 4(x^2 + y^2) \, dxdy$ on $D = \{ |z-1| \leq 1\}$. Seems like using polar coordinates is an option but what would be the boundary for a shifted circle?
Mondayisgood
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How to calculate the integral $ \int_{-2}^{2} e^x\sqrt{4-x^2} \mathrm{d}\, x$?

How to calculate the following integral: $$ I=\int_{-2}^{2} e^x\sqrt{4-x^2} \,\mathrm{d} x = ? $$ I try to use the substitution $x=2\sin\theta$ with $\theta\in[-\pi/2,\pi/2]$, and I get $$ I=4\int_{-\pi/2}^{\pi/2} e^{2\sin\theta}…
MHMH
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How do I solve the following $\int_0^1\frac1{\sqrt{1+x^4}}\,\mathrm dx$

I'm trying to solve the following integral: $$\int_0^1 \frac{1}{\sqrt{1+x^4}} \,dx$$ Here's a link to the original question: https://photos.app.goo.gl/mfTe649pzgKHeUKy7 I've tried to substitute $x$ as $\tan t$ and then $2t$ as $u$. Eventually I end…
Kevin Andrew
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value of $ (\int^{\infty}_{-\infty}xf(x)dx)^2$

Given $$ \int^{\infty}_{-\infty}e^{tx}f(x)dx = \sin^{-1}\bigg(t-\sqrt{\frac{1}{2}}\bigg),$$ then what is the value of $$ \bigg(\int^{\infty}_{-\infty}xf(x)dx\bigg)^2\quad ?$$ What I…
jacky
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Evaluating the integral $\int_{1/3}^3 \frac{\tan^{-1}(x) } {1+x^2-x} \, dx$

Would like some help in evaluating the integral $$\int_{1/3}^3 \frac{\tan^{-1}x}{ 1-x+x^2} \, dx$$
Leila
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Solving the integral $\int_{0}^{\pi}{\log(\sin x)}\,\mathrm dx$

Solve this integral$$\int_{0}^{\pi}{\log(\sin x)}\,\mathrm dx$$ I have tried this by breaking the limit from $0$ to $\pi/2$ and $\pi/2$ to $\pi$, but I am unable to solve the 2nd part.
Atanu Saha
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Find the bounds of two functions containing an absolute value

We were given the two functions $f_1(x) = |x|$ and $f_2(x) = x^2-2$ of which we needed to find the surface of. My solution this far is: However, I don't know which bounds I have to use for my integral. When I plotted the graph on my graphic…
LLScheme
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Find f '($π\over2$) by solving integrals

Hello can you help me please with this problem about integrals, I don't know how to solve g(x), or, how can I find f '($π\over2$). Here is the problem Thanks in advance.
Alexander
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Can I solve this using p4 property?$\int_{0}^{\pi/2}\frac{\cos^2x}{{\cos^2x + 4\sin^2x}}dx$

$$\int_{0}^{\pi/2}\frac{\cos^2x}{{\cos^2x + 4\sin^2x}}dx$$ Can I solve this question using p4 of definite integrals
Mark Henry
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Greatest integer function and integration

I know that in greatest integer function, we have to divide this into limits where the value of function changes. But in case of fraction what happens is just confusing. Can anyone help? Like if the limit is from $- 3/5$ to $3/5$ to integrate […
Ankita A
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How to solve this definite integration $\int_0^L{\sqrt{1+\left(\frac{k}{P(x)}\right)^2}dx}$?

Is there a way to further simplify or solve this integral? $$\int_0^L{\sqrt{1+\left(\frac{k}{P(x)}\right)^2}dx}$$ where $k$, $L$ are constant; $P(x)$ is a function defined over $0$ to $L$. $P(x)$ takes the…
Zhang Ze
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How do you prove the integral of a positive function is also positive?

It seems straightforward, every time I draw a picture of a function that's always greater than zero, then the area under it is also always greater than zero. But then when I look at the definition of an integral, it's some convoluted summation that…
user608672
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