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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.

20559 questions
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Why is the line integral not coming the same using different parameters

I need to calculate the line integral of (x^2 - x*y)dy when the curve is y=x^2 from (-1,1) to (2,4) If I convert the x's into y ie x^2=y and x=(y)^1/2 and use the limits 1 to 4 I get -4.9 If I use x as a parameter and convert the y into x and…
user379001
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Definite Integrals: what's wrong with my solution?

The answer to this question was supposed to be $-\frac{\pi}4$ using integration by parts but I thought this substitution was sufficient to get the answer.
A_for_ Abacus
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Finding the integral $\int_0^{\infty}\frac{\text{exp}\left(j \nu x\right)}{(x+1)^2}\,dx$

I have this integral $$\psi_X(j\nu)=E_X\left[e^{j\nu X}\right]=\int_0^{\infty}\frac{\text{exp}\left(j \nu x\right)}{(x+1)^2}\,dx$$ where $X\in[0,\infty)$ is a random variable with PDF $f_X(x)=(x+1)^{-2}$ and $j=\sqrt{-1}$. How can I evaluate this…
EngDavid
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Integral of a function which has different derivative from left and right direction at a point on an interval

If $f(x)=|x|$, then we know $f(x)$ has a sharp point at $x=0$, and now I want to calculate the derivative of $f(x)$ on interval $[0,1]$, i.e., $$\int_{0}^{1} f '(x)\,dx$$ but $f'(x)$ is not defined on $0$, so I want to know whether the function of…
扁頭科學麵Kevin Lee
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Prove definite-integrals

So...Try it again. The last time I did task: Prove equation $\int_{a}^{b}f(x)dx = \int_{a}^{b}f(a+b-x)dx$ $a,b$ $are$ $ constants$ Solution: Left part of equation $\int_{a}^{b}f(x)dx = F(b) - F(a)$ Right part of equation $\int_{a}^{b}f(a+b-x)dx…
Boris Molostvov
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Two complex definite Integrals

I want to solve these Integrals 1. $$\int_{0}^{\frac{\pi}{2}}\frac{1}{1+\tan^{\sqrt {2}}x} dx$$ 2. $$\int_{0}^{\frac{\pi}{2}}\frac{1}{(\sqrt{2}\cos^2x+\sin^2x)^2}$$ Thanks
Mani
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Can I solve this integral problem?

I want to know how can i solve this function. $\int{(1-y^d)^n}dy$ Is it possible to solve it? If you know the method, please teach me.
S. Jun
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Alternate Definition of Definite Integral

I was trying to solve this given problem, When $f(x)$ is continuous on $[a, b]$, there exists infinitely many reals $p_1, p_2, p_3$ and $q_1, q_2, q_3, q_4$, which satisfies the following equations. $$(1) \int_a^bf(x)dx=\lim_{n \to…
zxcvber
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Proving lemma on integrabilty

Let $f:[a,b]\to\mathbb{R}$ be a bounded function. I want to prove that if there exist sequences of tagged partitions $(P_n)_{n=1}^{\infty}$ and $(Q_n)_{n=1}^{\infty}$ of the interval $[a,b]$, so that: $$\inf_{n\geq 1} U(f,P_n)=\sup_{n\geq…
nono
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A misleading integration step.

Compute $$\int_{-\pi/2}^{\pi/2} \frac{x^2\cos(x)}{1+e^x}$$ The first step itself given in the solution is changing $e^x$ to $1/e^x$ . Now as first step is making no sense to me so I didn't post the whole solution. Hope you guys help and tell me…
Archis Welankar
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Closed Form for the Definite integral of the Modified Bessel Function of the First Kind?

Can anyone help me on deriving the closed form expression for the following definite integral of the modified Bessel function of the first kind, zero order, involving power and exponential functions $$I={\displaystyle\int\limits_0^{a } {{x^{b}}…
junxv penn
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$\int_a^b g(x) x dx < \int_b^c g(x) x \mathop{\mathrm{d}x}$ for $a < b < c$

Assume that $$\int_a^b g(x) \mathop{\mathrm{d}x} = \int_b^c g(x) \mathop{\mathrm{d}x}$$ for $a < b < c$. Does this imply that $$\int_a^b g(x) x \mathop{\mathrm{d}x} < \int_b^c g(x) x \mathop{\mathrm{d}x}$$ ? I strongly feel so but don't know how to…
bonifaz
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Find $\int_{-5}^{-3} \frac{dx}{\sqrt{x^2 - 4}}$ using trig substitution

$$\int_{-5}^{-3} \frac{dx}{\sqrt{x^2 - 4}}\\ x = 2\sec\theta \\ dx = 2\sec\theta \tan\theta \,d\theta\\ \theta \in[0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi]\\ $$ $$\int_{sec^{-1}\frac{-5}{2}}^{\sec^{-1}\frac{-3}{2}} \frac{2\sec\theta \tan\theta…
tildawn28
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Sum of infinite circles on a plane

A power $P_0$ is spread on infinite circles of radius $r_0$ so the incident power on every circle is $dP_0$. The center of every circle is located on a plane $\Gamma =\left(0\le x\le x_0,0\le y\le y_0\right)$. I am stuck in find the distribution of…
Riccardo.Alestra
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How do I integrate this exponential function?

$H = \frac{V^2}{R} e^{-2t/RC} dt$ From 0 to ∞. I put $k = e^{-2t/RC}$ I tried taking log on both sides but got, something like $log k - t/2k = -t/2RC$ How can I solve it
Anubhav Goel
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