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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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Evaluation of $\int_0^1 \frac{\ln(1-x+x^2)}{x(1-x)}dx$

To add one more solution link to this answer listing Feynman's trick exercises, I'm posting herein a calculation by said technique of $I_{12}:=\int_0^1\frac{\ln(1-x+x^2)dx}{x(1-x)}$,…
J.G.
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Wrong Question in Thomas Calculus book?

Here is the problem from Thomas Calculus book that I'm working on: Find $\displaystyle f\left(\frac{\pi}{2}\right)$ from the following information: $f$ is positive and continuous The area under the curve $y=f(x)$ from $x=0$ to $x=a$…
Abhishek A Udupa
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Evaluating $\int_0^{\infty} \frac{\ln (x)}{1-x^n}\text{d}x$

This is just for recreational purposes. I've been wondering how to evaluate the following integral: $$\int_0^{\infty} \frac{\ln (x)}{1-x^n}\text{d}x$$ Because I noticed that certain values for $n$ lead to some nice rational multiples of $\pi^2$ when…
Moni145
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Why $\int_0^{2\pi} \frac{dx}{2-\sin x}=0$?

Of course this is not true, because the integrand is always greater than zero. However, when using Mathematica to do an indefinite integral of the general form $$ \int\frac{dx}{\xi-\sin x} =…
an offer can't refuse
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Evaluating $\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$

I want to evaluate this integral $$\int_0^{2\pi} \sqrt{1+\cos^2(x)}\ dx$$ But I cannot find a useful substitution/strategy. Could you please give me a hint? I was thinking proving that this is equal to $$\int_0^{2\pi} \sqrt{1+\sin^2(x)}\ dx$$ but…
mez
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Difficult to evaluate $\int_{-\infty}^{\infty}\frac{x(x+a)(x+b)(x+c)}{(x^3+2x^2-x-1)^2+(x^2+x-1)^2}dx$

Where $a,b$ and $c$ are consecutive arithmetic terms. We wish to evaluate this integral, $$\int_{-\infty}^{\infty}\frac{x(x+a)(x+b)(x+c)}{(x^3+2x^2-x-1)^2+(x^2+x-1)^2}\mathrm dx$$ I don't even really know how to make an attempt. If I expanded the…
user569129
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Can't solve $\int_{0}^{\pi} \frac{x}{1 + \cos^2x} dx$

I tried this :- Let $$I =\int_{0}^{\pi}\frac{x}{1 + \cos^2x}dx\tag{1}$$ then $$I = \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2(\pi-x)}dx= \int_{0}^{\pi}\frac{\pi-x}{1 + \cos^2x}dx\tag{2}$$ Adding (1) and (2), we get $$ 2I = \int_{0}^\pi\frac{\pi}{1 +…
anonymous
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Evaluate $\int_0^1 \log \left( \frac{x^2+\sqrt{3}x+1}{x^2-\sqrt{3}x+1} \right) \frac{dx}{x} $

I'm trying to evaluate this integral but i'm having a lot of problems with the standards method. $$\int_0^1 \log \left( \frac{x^2+\sqrt{3}x+1}{x^2-\sqrt{3}x+1} \right) \frac{dx}{x} $$ I've tried integration by parts, taylor expansion of the $\log$…
Fabio
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Closed forms of $\int_0^\frac{\pi}{2}\frac{\sin(\alpha\phi)}{\sin(\phi)}d\phi$ and $\int_0^\frac{\pi}{2}\frac{\cos(\theta)}{\cos(\beta\theta)}d\theta$

I have recently looked into the two integrals:$$S(\alpha)=\int_0^\frac{\pi}{2}\frac{\sin(\alpha\phi)}{\sin(\phi)}d\phi$$ $$C(\beta)=\int_0^\frac{\pi}{2}\frac{\cos(\theta)}{\cos(\beta\theta)}d\theta, -1\le\beta\le1$$ I noticed that…
aleden
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Multiple answers in evaluation of a definite integral.

Consider this definite integral $$I=\int_0^{2\pi}\dfrac{x\cos x}{1 +\cos x}dx$$ Method 1 $$I=\int_0^{2\pi}\dfrac{x\cos x}{1 +\cos x}dx.....(1)$$ $$I=\int_0^{2\pi}\dfrac{(2\pi-x)\cos (2\pi-x)}{1 +\cos (2\pi-x)}dx.....(2)$$ Adding (1) and…
user585765
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Definite integral of $x\sin^n x$ from $0$ to $\pi/2$

How to find \begin{equation*} \int_0^{\pi/2} x\sin^n x dx \end{equation*} where $n$ is a positive integer? I tried $y=x-\pi/4$ and that gives \begin{equation*} \frac{1}{2^{n/2}}\frac{\pi}{4}\int_{-\pi/4}^{\pi/4} (\sin y+\cos y)^n…
Haoran Chen
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Solving or approximating infinitely nested integral

Let $f$ given by $$f(x) = g(x) + \int_0^x\left(g(x_1) + \int_0^{x_1} \left( g(x_2) + \int_0^{x_2} \ldots d_{x_n} \ldots \right) d_{x_2} \right) d_{x_1}$$ where $n \rightarrow \infty$ and $g(x)$ is strictly decreasing in $x$. How can such an integral…
bonifaz
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Find the definite integral

Find the definite integral $$\int_{1}^{2}\frac{(3x-1)(2x+3)}{x} dx$$ I have come to an answer of $16 - \ln(8)$ which I think is very wrong.. First used integration by parts
joe
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How to compute $\int_0^1 \frac{1}{1+x^4}\;dx$?

I am having problems trying to compute $$\int_0^1 \frac{1}{1+x^4}\;dx$$ Wolfram alpha gives an answer $$\frac{\pi + 2 \coth^{-1}(\sqrt{2})}{4 \sqrt{2}}$$
user66307
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Integral of exponential $\int_0^{+\infty} \exp\left(-a^2 x\left(\frac{x-6}{x-2}\right)^2\right) \frac{dx}{\sqrt{x}}$

Calculate the following integral $$\int_0^{+\infty} \exp\left(-a^2 x\left(\dfrac{x-6}{x-2}\right)^2\right) \dfrac{dx}{\sqrt{x}}$$ I think the relation between this integral and function gamma is clear, But I do not know what variables change I do.…
pablocn_
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