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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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Integral $\int_0^{\pi}\frac{1}{1+t\cos x} dx$

I figured out to use substitution $u=\tan\frac{x}{2}$ and arrived at integrate $\int_0^{\infty} \frac{2}{1+u^2+t(1-u^2)}du$ but am stuck here. Appreciate it if someone can drop some hint on how to proceed. Thank you.
Anonymous123
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How to solve $\int_{0}^{1} \frac{(x-1)\ln x}{(x+1)(x^2+2\cosh \alpha+1)} \,dx$

I have this unpleasant integral that appear in my last exam which i was not able solve For $\alpha \in \mathbb{R}$. Evaluate $$I=\int_{0}^{1} \frac{(x-1)\ln x}{(x+1)(x^2+2\cosh \alpha+1)} \,dx$$ Anyway my attempt is that I try to do some partial…
user635988
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Is the ordering of integrals important in a 2D integral?

Let's consider $\int_a^b\int_c^d f(x,y)dxdy$ : do we know that $x$ will vary from $a$ to $b$ and $y$ from $c$ to $d$, or said similarly, is the ordering crucial in the definition ? Same question for the ordering of dx and dy ? I'm not asking about…
Mathieu Krisztian
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one over square root x squared y squared integrated over a rectangle

What is the integral of $1/\sqrt{x^2+y^2}$ over a rectangle, i.e. what is the solution to the following integral? $$\int_{x_1}^{x_2} \int_{y_1}^{y_2} \frac{1}{\sqrt{x^2+y^2}} \, dy\, dx$$ I am aware that some posts indicate a change to polar…
Tim
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Problem involving mean value theorem for integrals

I am trying to solve the following question: $\def\pih{{\frac{\pi}2}}$ Does there exist $\xi \in [\pih,\pi]$ such that $\int_{\frac{\pi}2}^{\pi}\sqrt{1+x^2}\cos{x}dx = -\sqrt{1+\xi^2}$ ? Prove the result and state all theorems used. While trying…
Đumić Branislav
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Is this result of definite integral as the limit of a sum correct?

I know that, $$\int_{a}^b f(x) \, \mathrm{d}x = \lim\limits_{n \to ∞} \dfrac{b-a}{n} \sum\limits_{r=1}^n f \left( a + r\left(\dfrac{b-a}{n}\right) \right)$$ Then using that, does the following result hold? $$\lim\limits_{n \to ∞} \dfrac{1}{n}…
William
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Find the positive value of $k$ for which the value of the definite integral $\int_0^{\frac{\pi}2}|\cos x-kx|dx$ is minimized

Find the positive value of $k$ for which the value of the definite integral $\int_0^{\frac{\pi}2}|\cos x-kx|dx$ is minimized. If I draw the graph of $\cos x$ and the straight line $kx$, they intersect between $0$ to $\frac{\pi}2$. Let the abscissa…
aarbee
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Find $f(x)$ if $f(x)=A\int_0^{\frac{\pi}2}(\sin x\cos t\cdot f(t))dt+\sin x$, where $A$ is a constant

Find $f(x)$ if $f(x)=A\int_0^{\frac{\pi}2}(\sin x\cos t\cdot f(t))dt+\sin x$, where $A$ is a constant. $$f(x)=(A\int_0^{\frac{\pi}2}(\cos t\cdot f(t))dt+1)\sin x\\=B\sin x,$$ where $B=A\int_0^{\frac{\pi}2}(\cos t\cdot f(t))dt+1$ .... $(1)$ Also,…
aarbee
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Integrating Euler's function

Let $\phi(x)$ be the Euler function. As presented in the Special values section there, Using the Pentagonal number theorem, exchanging sum and integral, and then invoking complex-analytic methods, one derives $$\int_0^1 \phi(x) \, dx = …
fox
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Evaluating $ \int_{-1}^{1} \arctan \left(e^{x}\right) d x $

Using the substitution x=ln(t), the integral becomes $$ I=\int_{e^{-1}}^{e} \frac{\operatorname{Arctan}(t)}{t} \cdot d t $$ which has no antiderivative expressed with usual functions. First, i want to prove that $$\frac{\operatorname{Arctan}(t)}{t}…
Yagami
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integration by parts for numerical definite integrals

I am new to calculus, and I am trying to learn integration by parts. So I have got a question here, and I have no idea what I am doing wrong (i have been looking at it for about 4 hours now..) The method as I understand it, is as follows: $\int f(x)…
mbih
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Need help with integral and change the variable

I have $$\displaystyle \int_0^\pi \sqrt{x} ~\cos x ~dx$$ and I need to make change of the variable $u = \sin x$.
user79727
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Are limits of integration defined with respect to the pronumeral or the differential?

Consider the indefinite integral $$\int x^2 \,d(x^2).$$ It evaluates fairly easily to $\tfrac{x^4}{2} + C$. My question is about what happens when we start evaluating definite integrals with respect to these functions. In a specific example, how…
L.Coy
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Integral of $\sin(x)^{\frac{4}{3}}$

How does one solve the integral $\int_{0}^{2\pi}(\sin(x))^{4/3}\ \mathrm{d}x$, or more generally $\int_{0}^{2\pi}(\sin(x))^{(m+n)/m}\ \mathrm{d}x$ ? Standard CAD math as Maple did not provide answers.
pivu0
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Defining area between three curves

The area bounded by the circle $x^2 +y^2=8$ , the parabola $x^2 =2y$, and the line $y=x$ is..? The integration is pretty easy to do but I'm having a difficult time finding the area between three curves in the above figure. What would you define…
tryst with freedom
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