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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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Calculate the definite integral:

The integral was: $$\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\frac{\pi + 4x^6}{1-\sin(|x|+\frac{\pi}{6})}$$ What I did was to identify that its an even function and write it as: $$2\int_{0}^{\frac{\pi}{6}}\frac{\pi +…
infinite-blank-
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Solve for $\int_{0}^{+\infty} \frac{\ln \left(x^{2}+2 \sin a \cdot x+1\right)}{1+x^{2}} d x$

It is known that $$\int_{0}^{+\infty} \frac{\ln \left(x^{2}+2 \sin a \cdot x+1\right)}{1+x^{2}} d x=\pi \ln \left|2 \cos \frac{a}{2}\right|+a \ln \left|\tan \frac{a}{2}\right|+2 \sum_{k=0}^{+\infty} \frac{\sin (2 k+1) a}{(2 k+1)^{2}}$$ In an attempt…
Leo
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Integral involving nested Log

I've been trying to solve this integral for a few days. $$\int_0^{\infty}\left(\frac{1}{n}\left(t+n\right)\ln\left(\frac{t+n}{t}\right)-\ln\left(\frac{1}{n}\left(t+n\right)\ln\left(\frac{t+n}{t}\right)\right)-1\right)dt$$ For $n\gt0$. I'm able to…
tyobrien
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Solving a definite integral from zero to infinity:

Find the value of: $ \int _{0}^{ \infty} \frac{ \ln x}{x^2+2x+4}\,\text{d}x$ Here I factorised the denominator into complex factors, and performing partial fraction decomposition, I get the following integral I cannot solve: $\int…
user638473
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The integral $\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}=a\pi,~ a, b \in \Re ?$

This integral $$\int_{-\infty}^{\infty}\frac{ a^2x^2 dx}{(x^2-b^2)^2+a^2x^2}=a\pi, ~ a, b \in \Re$$ looks suspiciously interesting as it is independent of the parameter $b$. The question is: What is the best way of proving or disproving this?
Z Ahmed
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Double integral computation with triangular function

Suppose we have $6$ constants, $(x_l, x_m, x_u$, $y_l, y_m, y_u)\in \mathbb R^6$ such as: $$x_l < x_m < x_u$$ $$y_l < y_m < y_u$$ Let $D = \left\{(x, y) : x_l\le x\le x_u, \quad y_l \le y \le y_u, \quad y < x\right\}$ I want to…
JKHA
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Evaluate $\int_0^{\frac{\pi}{4}}\arctan\frac{\sqrt{2}\cos 3x}{(3+2\cos 2x)\sqrt{\cos 2x}}{\rm d}x$.

Problem Evaluate $$ \int_0^{\frac{\pi}{4}}\arctan\frac{\sqrt{2}\cos 3x}{(3+2\cos 2x)\sqrt{\cos 2x}}{\rm d}x.$$ The integrand is complicated. AlphaWolfram outputs the result is zero. If we consider making a subsitution, let $x=\dfrac{\pi}{4}-u$,…
mengdie1982
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is $\int_{-1}^{1} \frac{2x}{x^2 - 9}dx = 0$?

am I doing this the right way? $\int_{-1}^{1} \frac{2x}{x^2 - 9}dx = \int_{-1}^{1} \frac{d(x^2 - 9)}{x^2 - 9}dx = \left[\ln|x^2 - 9|\right]_{-1}^{1} = 0$
Hamza
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Evaluate the integral: $\int_0^{\infty}\frac{\tan^{-1}(tx)}{x\left(1+x^2\right)} \mathrm{d}x$

I've been trying to evaluate the integral for a while now, and I've been unable to find it anywhere... I tried substituting $\tan^{-1}(tx)$ as $u$ but got nowhere... I have done dozens of other substitutions... I've been told to use the properties…
PCeltide
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Why is $\int_{0}^{1} \log x $ computable?

While evaluating $$\lim_{n \to \infty}\left(\frac{n!}{n}\right)^{1/n} $$ The integral turns out to be $$\int_0^1 \log x = \big[x\log x\big]_0^1 - \big[x \big]_0^1 $$ The second term will -1 how ever the first term will be $$I_1=1\times\log 1…
user619072
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How is this derived?

In my textbook I find the following derivation: $$ \displaystyle \lim _{n \to \infty} \dfrac{1}{n} \displaystyle \sum ^n _{k=1} \dfrac{1}{1 + k/n} = \displaystyle \int^1_0 \dfrac{dx}{1+x}$$ I understand that it's $\displaystyle \int^1_0$ but I don't…
Homework
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Can anyone help in $\int_0^\pi (\sin x )^{\cos x} dx$?

I am trying to solve the following definite integral; $\int_0^\pi (\sin x )^{\cos x} dx$?
Albert
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Not able to integrate

$\displaystyle\int_{0}^{\pi/2} \frac{\sin\left(x\right)} {\left[1 + \,\sqrt{\,\sin\left(2x\right)\,}\,\right]^{2}} \,\mathrm{d}x$ i used the property to change reach $\displaystyle 2I = \int_0^{\pi/2}\frac{\sin\left(x\right) +…
maveric
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Upper bound for $\displaystyle\int_1^\infty \frac{\lfloor x\rfloor }{x^3} dx $

How do I find $f(x) > \lfloor x \rfloor $ in the interval $[1, \infty)$ such that $$\int_1^\infty \frac{2\lfloor x\rfloor }{x^3 }dx < \int_1^\infty \frac{2f(x)}{x^3} dx = \frac {10}{6}$$ Actually I am trying to show that $\large\frac {10}{6}$ is…
S L
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Finding $\int^{\pi}_{0}\sin(8x+8\sin 3x)dx$

Finding $\displaystyle \int^{\pi}_{0}\cos^4(x+\sin 3x)dx$ Try: From $\displaystyle \cos^4(x)=(\cos^2(x))^2=\frac{1}{4}\bigg[1+\cos 2x\bigg]^2$ $$=\frac{1}{4}+\frac{1}{8}\bigg(1+\cos^2(4x)+2\cos 4x\bigg)+\frac{\cos…
DXT
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