Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [integration]

For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.

Integration is a major part of .

There are two main kinds of integrals:

  • definite integrals (e.g. proper and improper integrals), which often have numerical values
  • indefinite integrals, which group families of functions with the same derivative.

Several techniques to solve integrals have been developed, including integration by parts, substitution, trigonometric substitution, and partial fractions.

Integration can be used to find the area under a graph and find the average of the function. Also, it can be used to compute the volume of certain solids and to find the displacement of a particle.

73636 questions
8
votes
2 answers

Why does the definition of an integral specify a closed interval?

Here's the definition of an integral from Wikipedia: Given a function $f$ of a real variable $x$ and an interval $[a, b]$ of the real line, the definite integral $$\int_a^b f(x) \, dx$$ is defined informally to be the area of the region in the…
Jim_CS
  • 5,331
8
votes
3 answers

Evaluation of $ \int_{0}^{\pi}\ln(5-4\cos x)\,dx$

Evaluation of $\displaystyle \int_{0}^{\pi}\ln(5-4\cos x)dx = \int_{0}^{\pi}\ln(5+4\cos x)dx$ $\bf{My\; Try::}$ Let $\displaystyle I(a,b) = \int_{0}^{\pi}\ln(a+b\cos x)dx$ Then $$\frac{d}{db}(a,b) = \frac{d}{db}\left[\int_{0}^{\pi}\ln(a+b\cos…
juantheron
  • 53,015
8
votes
3 answers

Is there an elegant way to solve $\int \frac{(\sin^2(x)\cdot \cos(x))}{\sin(x)+\cos(x)}dx$?

The integral is: $$\int \frac{(\sin^2(x)\cdot \cos(x))}{\sin(x)+\cos(x)}dx$$ I used weierstraß substitution $$t:=\tan(\frac{x}{2})$$ $$\sin(x)=\frac{2t}{1+t^2}$$ $$\cos(x)=\frac{1-t^2}{1+t^2}$$ $$dx=\frac{2}{1+t^2}dt$$ Got this: $$\int…
marius
  • 145
  • 6
8
votes
2 answers

Geometric intuition of improper integrals

I am aware that the area under the curve of $\frac{1}{x}$ is infinite yet the area under the curve of $\frac{1}{x^2}$ is finite. Calculus and series wise, I understand what is going on, but I can't seem to get a good geometric intuition of the…
Frank
  • 133
  • 5
8
votes
2 answers

An integration question to be solved without using differentiation under the integral sign.

$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$ As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
Pablo
  • 81
8
votes
1 answer

Non-integrable function that has an antiderivative

The wikipedia article on antiderivatives states: Non-continuous functions can have antiderivatives. [...] In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions…
Zuza
  • 237
8
votes
1 answer

Show by substitution that $\int_0^{\pi} \frac{x\sin x}{1+\cos^2 x} \,\mathrm dx = \frac{\pi}{2}\int_0^{\pi} \frac{\sin x}{1+\cos^2 x} \,\mathrm dx$

How do you show $$\int_0^{\pi} \frac{x\sin x}{1+\cos^2 x} \,\mathrm{d}x = \frac{\pi}{2}\int_0^{\pi} \frac{\sin x}{1+\cos^2 x} \,\mathrm{d}x$$ without integrating by parts, but only using substitution?
Kirthi Raman
  • 7,564
  • 2
  • 36
  • 60
8
votes
1 answer

Integrating a rational function.

How to integrate $$\int_1^{\infty}\frac{2x^3-1}{x^6+2x^3+\sqrt3x^2+1}{\rm d}x$$ The bottom is not factorizable hence no partial fractions. There seems no other way.
RE60K
  • 17,716
8
votes
2 answers

Do regular partitions suffice for Riemann integrability of a real-valued function on a closed interval?

Does the following condition for a bounded function $f: [a, b] \to \mathbb R$ suffice that it be Riemann integrable on $[a,b]$, with Riemann integral $I$? For every $\epsilon > 0$, there exists a positive integer $N$ such that, for every positive…
murray
  • 778
  • 3
  • 16
8
votes
5 answers

What does integration do?

I know that integrals are used to compute the area under a curve. Let's say I have $y = x^2$. It creates smaller rectangles and then add up the sum (assuming that rectangles are going infinitely in number and is like going to a limit). But I…
vvavepacket
  • 533
7
votes
2 answers

Tricky integration/functions problem

For $x>0$, let $f(x) = \displaystyle \int_1^x\frac{\ln t}{1+t}dt$. Find the function $f(x) + f(1/x)$ and show that $f(e) + f(1/e) = 1/2$. Any help would be thoroughly appreciated.
user34304
  • 2,749
7
votes
6 answers

Find the integral $\int_{0}^{\frac{\pi}{4}}\frac{\sin{x}\cos{x}}{\sin{x}+\cos{x}}dx$

find the integeral $$\int_{0}^{\frac{\pi}{4}}\dfrac{\sin{x}\cos{x}}{\sin{x}+\cos{x}}dx$$ I know…
math110
  • 93,304
7
votes
3 answers

Integrate $\int \sqrt{(\sec{x} +\tan{x})}\ \cdot \sec^2x\,dx$

Integrate: $$\int \sqrt{(\sec{x} +\tan{x})}\ \cdot \sec^2x\,dx$$ My attempt : I substituted $\sec{x} + \tan{x} $ as $t^2$ Then, $$ (\sec{x} \cdot \tan{x} + \sec^2x) dx =2tdt$$ $$\sec{x}( \tan{x} + \sec{x}) dx =2tdt$$ $$\sec{x}\cdot t^2 dx…
maths lover
  • 3,344
7
votes
2 answers

Evaluating $\int\limits_{-\infty}^\infty {\exp(iax)\over1+ix}dx$

How does one evaluate the integral $\int\limits_{-\infty}^\infty {\exp(iax)\over1+ix}dx$? I tried Wolfram Alpha, but it just says "computation timed out"... I tried the indefinite integral and got an answer involving some weird function $E_1$. Is it…
owen
  • 79
7
votes
4 answers

What is the integral of $x(1-x)^8$?

I want to find the integral of $x (1-x)^8$. How do I go about this? For example, which rule do I use from http://integral-table.com ? Thanks!
nubela
  • 841
1 2 3
…
99 100