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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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Conflicting answers both claiming to be right

My book lists the answer as: http://www.chegg.com/homework-help/calculus-4th-edition-chapter-5.3-solutions-9780495557425 But I keep getting 5/3, and both Symbolab and Wolfram agree with my answer. Which answer is right?
user510634
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Use the definition of the integral to evaluate the integral.

$\int_0^1$ $(x^3-3x^2)$ dx The definition of the integral that must be used is $\int_a^b$ $f(x)$ dx = $\lim_{x\to\infty} \sum_{i=1}^n f(x_i)\Delta x $ where $\Delta x = \frac{b-a}{n}$ and $x_i=a+i \Delta x $ The answer is $\frac{-3}{4}$ and I am…
Katelyn
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Integral from odd & even function

Sorry this is my first time asking in forum because, please do critics how i ask question, give me some tips so I can be more clear to ask question. If : $\int_{-2}^2g(x)(f(x)+1)dx=8$ $\int_{-1}^2g(x)dx=5$ With : $f(x)$ is an odd function ($f(-x) =…
Chrystian N
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Area between curve and two lines

I have three functions: $$y=5+\sin x\\ y=-x+0.5\\ y=2x+1$$I have to calculate area between them. I made: $-x+0.5=2x+1$ and I got $x=-\frac16$. What shall I do next to get the area?
šm98
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Limit of integral of sequence of functions

Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be an integrable function. Find : $$\lim_{n \to \infty} n \int_0^1\frac{f(nx)}{1+x} \, dx$$ I've thought about using the dominated convergence theorem, but I can't seem to be able to satisfy its conditions for a…
Blencer
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Integration dependent on the ir/rationality of a,b

Let $$\text{I(a,b)}=\int _0^1\frac{\ln(1+x^a)}{1+x^b}$$ where $a=\text{irrational},b=\text{rational}$. Can this integral have a closed form ie definite value at some special $a's,b's$?
Archis Welankar
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Formula for integral, $\int\limits_{B(0,1)} x^a y^b z^c dx\ dy\ dz$, over unit ball

Is there any general formula for the integral $$\int\limits_{B(0,1)} x^a y^b z^c dx\ dy\ dz$$ where $B(0,1)$ is the unit ball centered at $x=y=z=0$ and $a,b,c$ are positive integers.
Jim
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Is there any reduction formula possible?

$$ I=\int_{0}^ {\pi} e^{x} (sinx)^{n} dx $$. I tried a lot but unable to obtain and stuck by it.
Girish Kumar Chandora
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Integral with unknown part

if I have an area between two curves, f(x)=x^2 and g(x)=x^(1/2) would there be a way to algebraically calculate 1/4 of the area and show it in two different parts of the area starting from the bottom intersect? enter image description here Hopefully…
flynn swift
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Definite integration involving log

Please help me to solve following definite integral: $$ \int_{0}^{1} {\sqrt[3]{x\log\frac{1}{x}}dx}$$
user458519
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Need help in solving defined integral problem.

The graph of the continuous function $y = f (x)$ is symmetric with respect to the origin, for all real numbers $x$ If $$ f(x)=\frac{\pi}{2}\int_1^{x+1}f(t)dt$$ and $f(1)=1$ Find $$\pi^2\int_0^1xf(x+1)dx$$ I have tried:$$f'(x)=\frac{\pi}{2}f(x+1)$$ …
R.Temur
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how to integrate 1-tanh(x)

I think that for solving $\int_0^\infty 1-tanh(x)dx$ I have to use the fact that $tanh(x)=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$, so that the integral becomes: $2\int_0^\infty \frac{e^{-x}}{e^{x}+e^{-x}}dx$ At this point I was thinking to change…
shamalaia
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Dependency on x? Definite integrals

Prove that the values of the $$\int_{-cos(x)}^{sin(x)} \frac{1}{\sqrt{1-t^2}}dt, x \in (0, \frac{\pi}{2})$$ Do not depend on x. I don't know what this means. I just found the derivative. $$\leftrightarrow -\int_{0}^{-cos(x)}…
Tinler
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Impossible integral appears when trying to solve $t^{\beta} u_{t} - u_{xx} = f(x,t)$.

I have done a reduction of this equation: $t^{\beta} u_{t} - u_{xx} = f(x,t)$ to an ODE and this integral seems to be impossible: $$\int_{0}^{T} e^{\frac{-2 \lambda t^{1- \beta}}{1- \beta}} dt$$ I've tried Wolfram, but it doesn't work either.
Danilog
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Two parametric integral

How to obtain the explicit formula of the following integral $$\int\limits_0^1 {{t^{ - 1 - x}}\left( {{{\left( {1 + t} \right)}^{ - y}} - 1} \right)} dt,\;x,y \in \left( { - 1,1} \right).$$ It can be expressed in terms of hypergeometric function？
xuce1234
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