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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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Integration using partial metod

Need help on solving integrals using partial integration. As I have only solved ones with Newton-Leibniz, I don't know how to solve this types: $$\int_0^{\pi/4} \frac{x\sin(x)}{\cos^2(x)}dx$$
nlimits
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Integrate via substitution

Need help on solving integrals using subsitution. As I have only solved ones with Newton-Leibniz, I don't know how to solve this types: $$ \int_0^2 \frac{dx}{\sqrt{x+1}+\sqrt{(x+1)^3}} dx$$
nlimits
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Using differentiation under the integral sign to evaluate a trigonometric function

I am trying to evaluate the following: $\int_0^\pi (5+3\cos(x))^{-3} \mathrm{d}x$ by using differentiation under the integral sign, however I cannot seem to find a suitable position to insert my '$A$' variable. I have tried substituting the power of…
Tech
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Find dy/dx of Integral

really stuck on this problem, my textbook doesn't have ANYTHING like it. The only instruction is to find the dy/dx of the interval: $$y=x\int_2^{x^2}\sin(t^3)\,\mathrm dt $$ Thanks for any help!
Joe Caraccio
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Calculating the definite integral of a function

I have a problem with understanding the following integral: $\int_{x_1}^{x_1+\Delta x}[{u(x_1+\Delta x,t)-u(x_1,t)]dx}=[u(x_1+\Delta x,t)-u(x_1,t)]\Delta x$. I don't understand why we can find the integral of u by just multiplying it by the…
David
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Definite integral $\int_{1}^3{\frac{dx}{x^2-6x+8}} $

How to continue? $$\int_{1}^3{\frac{dx}{x^2-6x+8}} =$$ $$ =\left[ \frac12 \ln\left|\frac{x-4}{x-2}\right| \right]_{1}^3 =$$ $$= \frac12 \ln\left|\frac{-1}{1} \right| - \frac12 \ln\left|\frac{-3}{-1} \right|$$ How to solve integrals like this, when…
DavidM
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How to find an equation that goes through two points, with a definite integral.

Trying to find proper variables for a drinking game wherein you drink 100 times over the course of 100 minutes, wherein the first drink is taken after 0.25 minutes (15 seconds) and the last one is taken after 2 minutes. I've approximated an…
DrLime2k10
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Can not resolve integral

I am doing an exercise about surface integral. I think that all my steps are good and that I am just blocking on the last step of the calculations, which is, if my steps are right, $$ \int_0^1 u^2\sqrt{1+4u^2}\;du $$ I don't figure out how to solve…
wayland700
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Moment of inertia of a cube through its diagonal

Show via direct integration of $ I = \int^M_0 r^2 dm $ that the moment of inertia of a cube, with side length $a$, and uniform density $\rho$, about an axis that passes through two opposite corners is $ I = \frac{ma^2}{6}. $ Setting up this…
Loonuh
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polynomial integral concept question: a term removed from polynomial, but integral is the same.

Sorry I have no better name for this question. while doing homework I came across this equation: And I am completely baffled on how this works. So can someone explain to me why they are equal because I simply do not see how they can be related in…
Spongebob Squareroot-pants
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How to express F(x) in terms of G(x)?

How to express $$F(x)=\int _{ 1 }^{ x }{ \frac { e^{ t } }{ t^2 } dt} $$ in terms of $$ G(x)=\int _{ 1 }^{ x }{ \frac { e^{ t } }{ t } dt} $$ ?
user220382
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Integral: arbitrary constant

I'm studying integrals and primitives and I have this enormous doubt about whether or not write and arbitrary constant $C$... And should I determine that constant. See an example: Discover the only diferenttiable function $f: \mathbb{R}…
Granger Obliviate
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Using integrals to discover area limited by 3 functions

The exercise is: calculate the area of the plane region limited by the curves $$y= x^2$$ $$y=x^2/2$$ $$y=x$$ I know that I need to use integrals and I know how to apply them, my only difficulty in this problem is that I'm having trouble by finding…
Granger Obliviate
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Finding the correct substitution for integral representaion

I have as a part of a problem the following integrals $\frac{1}{T}\int_{t-T}^t f(\tau) \, \mathrm{d}\tau = \frac{1}{T}\int_{0}^T f(t-u) \, \mathrm{d}u$, where $T > 0$ and $u \in [0,T]$ I cannot find the right substition. Can someone provide a hint?
Carlos
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Triple integral problem

$$\int_0^1 \int_0^z \int_y^z z\sin(x^2) \,\mathrm dx\,\mathrm dy\,\mathrm dz$$ I don't have any idea about how to graph given region. to me, intuitively this is not easy to imagine. somebody can analyze this integral?
fbg
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