Mathematics Stack Exchange
Mathematics Stack Exchange

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.

20559 questions
0
votes
3 answers

Evaluate $\int_{0}^{1}\frac{\sin[a\ln(1-x)]}{\ln(1-x)}\cdot x\mathrm dx$

How do we evaluate $$I=\int_{0}^{1}\frac{\sin[a\ln(1-x)]}{\ln(1-x)}\cdot x\mathrm dx$$ Making a sub: $u=\ln(1-x)$ this lead to a messy integral. I believe the closed form for $I=\arctan\left(\frac{a}{a^2+2}\right)$, how to shows it, I am not sure.…
user569129
0
votes
6 answers

Evaluate $\int_0^{\infty}\frac{\sqrt{x}}{x^2+1}dx$ without Complex Analysis

I found this question on a Complex Analysis Qualifying Exam: How would you evaluate: $$\int_0^{\infty}\frac{\sqrt{x}}{x^2+1}dx$$ I am interested in all methods including Complex Analysis, I'm just less familiar with it. For instance, since the…
user689775
0
votes
2 answers

Area enclosed between the curves

Find the area enclosed between $y=1/2$ and $ y= \cos x $ for $x =[0,2π]$ I am a bit confused with this question because there is no area bound between these graphs and a book suggests to enclose areas using $Y$ axis which I find is a bit incorrect…
Rishabh Shetty
  • 23
  • 7
0
votes
1 answer

How does one prove the following integration formula

$$\int_0^{\infty}\frac{\cos(z)\exp(-w z^2)-1}{z^2}dz=-\sqrt{\pi w}\,\exp\left(-\frac{1}{4w}\right)-\frac{\pi}{2}\,\operatorname{erf}\left(\frac{1}{2\sqrt{w}}\right)$$where w is positive and erf is the error function.
Honza
  • 383
0
votes
1 answer

Strange integral, $\int_{0}^{1}(-1)^xdx$

How do I evaluate this integral? $$\int_{0}^{1}(-1)^x \mathrm dx$$ Do I need to use $e^{i\theta}=\cos(\theta)+i\sin(\theta)$
user569129
0
votes
4 answers

variable of integration not in function

what is $\int_0^x ax^2 dy$ additionally, how is this described when the variable of integration is not in the function? I'm not sure how to research this.
rocketboy2000
  • 39
0
votes
1 answer

Show that $\int_{-1}^{1} \frac{1-x^2-x^3-x^{30}}{\sqrt[5]{(7-5x^2-x^{30})^4}}~dx<\frac{1}{3}$

By evaluating the integral or otherwise can one show by hand that $$\int_{-1}^{1} \frac{1-x^2-x^3-x^{30}}{\sqrt[5]{(7-5x^2-x^{30})^4}}~dx<\frac{1}{3}?$$
Z Ahmed
  • 43,235
0
votes
3 answers

Finding $\int^{\infty}_{0}\bigg(\frac{1-\cos 7x}{x}\bigg)e^{-x}dx$

Finding $\displaystyle \int^{\infty}_{0}\bigg(\frac{1-\cos 7x}{x}\bigg)e^{-x}dx$ Plan $$I =\int^{\infty}_{0}\frac{1}{x}\bigg(1-\frac{(7x)^2}{2!}+\frac{(7x)^4}{4!}+\cdots \bigg)\bigg(1-\frac{x}{1!}+\frac{x^2}{2!}+\cdots \bigg)dx$$ How do i solve it…
jacky
  • 5,194
0
votes
3 answers

Show that $\int_{0}^{\sqrt{3}} \sin^{-1} \left( \frac{2x}{1+x^2} \right) ~dx =\frac{\pi}{\sqrt{3}}.$

Show that $$\int_{0}^{\sqrt{3}} \sin^{-1} \left( \frac{2x}{1+x^2} \right) ~dx =\frac{\pi}{\sqrt{3}}.$$ When I do the following integral by parts taking $\sin^{-1}()$ as first function and 1 as second, I get an additional log…
Pallavi
  • 442
  • 3
  • 18
0
votes
1 answer

Theorem about monotone functions integration, a case when it is strictly augmenting function

In the proof of the theorem, it is supposed, that for a function, which is monotone and integrable on the interval $[a,b],\ f(a) < f(b)$, and $\delta$ is chosen to be equal to $\frac{1}{2}\frac{\epsilon}{f(b)-f(a)}$ Then the question is whether…
user
  • 1,412
0
votes
0 answers

Why there exist definite integral for a constant function $f(x)$?

For a proof of a existence of definite integral for a monotone function it is noted that if a function is a constant then it has a definite integral. Why?
user
  • 1,412
0
votes
1 answer

complex irrational definite integration

Finding value of $$\frac{8}{\pi}\int^{2}_{0}\frac{1}{x^2-x+1+\sqrt{x^4-4x^3+9x^2-10x+5}}$$ Let $$I=\int^{2}_{0}\frac{1}{x^2-x+1+\sqrt{x^4-4x^3+9x^2-10x+5}}dx$$ $$I=\int^{2}_{0}\frac{1}{x(x-1)+\sqrt{(x^2-2x)^2+5x(x-2)+5)}}dx$$ From $\displaystyle…
jacky
  • 5,194
0
votes
1 answer

What is the value of $I^2$ if $I=\int^{\pi/4}_{-\pi/4}\frac{\sqrt{\cos x}}{\left(\sqrt{\cos x}+\sqrt{\sin x}\right)^5}\,dx$?

If $$I=\int^{\pi/4}_{-\pi/4}\frac{\sqrt{\cos x}}{\left(\sqrt{\cos x}+\sqrt{\sin x}\right)^5}\,dx$$, then what is $27I^2$? Plan: $$I=\int^{\frac{\pi}{4}}_{-\frac{\pi}{4}}\frac{\sec^2 x}{\left(1+\sqrt{\tan x}\right)^5}\,dx$$ Put $\tan x=t$ and…
jacky
  • 5,194
0
votes
1 answer

Prove $\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$

Problem Prove $$\sqrt{1-e^{-1}}<\frac{1}{\sqrt{\pi}}\int^1_0 e^{-x^2}dx<\sqrt{1-e^{-2}}.$$ Attempt Since $$ \forall x\in[0,1],\quad e^{-x^2}=\sum_{n\geq 0}\frac{(-1)^n}{n!}\,x^{2n}$$ and by integrating termwise over $[0,1]$ we get: $$…
mengdie1982
  • 13,840
  • 1
  • 14
  • 39
0
votes
3 answers

If $I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is

If $\displaystyle I_{n}=\int^{1}_{0}x^2(1-x^2)^ndx,$ Then $\displaystyle \lim_{n\rightarrow \infty}\frac{I_{n+1}}{I_{n}}$ is plan $$I_{n}=-x\cdot…
jacky
  • 5,194
1 2 3
…
76 77