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
7
votes
1 answer

how to find $\int_{0}^{1}h_n(x)dx?$

I would appreciate if somebody could help me with the following problem: $$h(x)=\begin{cases} 2x&\left(0\leq x\leq \frac{1}{3}\right)\\ \frac{1}{2}x+\frac{1}{2} &\left(\frac{1}{3}
Young
  • 5,492
7
votes
4 answers

Finding $\int^{\frac{\pi}{2}}_{0}\ln(\sin x)\cdot \sin xdx$

Finding $\displaystyle \int^{\frac{\pi}{2}}_{0}\ln(\sin x)\cdot \sin xdx$ What I try:-> Integration by parts assuming $\displaystyle I = \int\ln(\sin x)\cdot \sin xdx = -\ln(\sin x)\cdot \cos x+\int\frac{\cos^2 x}{\sin x}dx$ $\displaystyle I =…
DXT
  • 11,241
7
votes
1 answer

Is the integral less than $\frac{n!}2?$

Knowing that $$\int_0^\infty x^n\exp(-x)\,dx=n!,$$ can we prove the following inequality: $$\int_0^n x^n\exp(-x) \,dx< \frac{n!}2 \;\;?$$
Aymane Gr
  • 187
7
votes
4 answers

How to integrate $\int_1^e \ln{x} \, dx$

This seems really tricky to me. I can't figure out how to integrate $\ln x$.
xyres
  • 309
7
votes
2 answers

How to calculate $\int_0^\infty \frac{1}{(1+x^2)(1+x^{2018})}\,dx$?

$$\int_0^\infty \frac{1}{(1+x^2)(1+x^{2018})}\,dx$$ My Calculus professor asked a challenge problem to one of my friends and asked her to evaluate it. I tried partial fractions to no avail and the trig substitution $x = \tan\theta$, but that leaves…
WhatsDUI
  • 570
  • 1
  • 5
  • 18
7
votes
1 answer

Can unbounded functions be Riemann integrable?

I need to prove or disprove that the next function is Riemann integrable on $[0,2]$: $$ f(x) = \begin{cases} \dfrac{1}{x} &x > 0 \\ 0 &x = 0 \end{cases} $$ My intuition is that it's not, because $f$ is unbounded in that interval, so $U(f,Pn) =…
McLovin
  • 550
7
votes
0 answers

formula that can be used to integrate powers of log-sin

I found a formula which, upon differentiating, can be used to evaluate various powers of log-sine or log-cos integrals. $\displaystyle…
Cody
  • 14,054
7
votes
2 answers

Different answers using different methods

I stumbled upon this integral $$\int_0^{2\pi} \frac{4ie^{i\theta}}{4e^{i\theta}} d\theta$$. Now if if do it using the formula$$\int\frac{f'(x)}{f(x)} dx=ln|f(x)|+C$$ then i have $$=ln|4e^{2\pi i}|-ln|4e^{(0) i}|$$ $$=ln|4|-ln|4|$$ $$=0$$ But if i…
Ayan Shah
  • 491
7
votes
2 answers

How can I evaluate $\displaystyle\int_{0}^{\infty}\frac{x\log(x)}{1+e^x}\,dx$?

My attempt : Evaluate a more general case, $$F(a) = \int_{0}^{\infty}\frac{x\log x}{1+e^x}\cdot a^{1+e^x} \,dx$$ $$F'(a) = \int_{0}^{\infty}x\cdot \log(x)\cdot a^{e^x} \,dx$$ Is there any way to take it from here without using by parts multiple…
Mathoholic99
  • 316
7
votes
2 answers

Integral $\int_0^\pi \big( (1+\alpha \cos x) \cos x \big)^n dx $

I have been struggling with the integral $$ I_n(\alpha) = \frac{1}{\pi} \int_0^\pi \big( (1+\alpha \cos x) \cos x \big)^n dx,$$ where $\alpha$ is real and $n$ is a non-negative integer. It is relatively easy to get the values for specific…
z.v.
  • 652
7
votes
1 answer

Evaluate $\int_0^1 \sin\left(\sqrt{\frac{1-x}{x}}-\sqrt{\frac{x}{1-x}}\right)\frac{dx}{x\left(3x^2 - 3x +1\right)}$

I have it on good authority that the following monstrosity $$I=\int_0^1 \sin\left(\sqrt{\frac{1-x}{x}}-\sqrt{\frac{x}{1-x}}\right)\frac{dx}{x\left(3x^2 - 3x +1\right)}$$ is not only convergent but has an analytic closed form. After spending a long…
user1892304
  • 2,808
7
votes
1 answer

A student forgot the Product Rule for differentiation and made the mistake of thinking that $(fg)'=f'g'$...

A student forgot the Product Rule for differentiation and made the mistake of thinking that $(fg)'=f'g'$. However, he was lucky and got the correct answer. The function f that he used was $f(x)=e^{x^2}$ and the domain of his problem was…
bigfocalchord
  • 5,597
7
votes
3 answers

How to integrate $\frac{1}{2^{(\ln x)}}$?

How to integrate $$\frac{1}{2^{(\ln x)}}$$? I tried using substitution $u=\ln x$, but $du=\frac{dx}{x}$ is not in the original equation.
Anne L
  • 71
7
votes
2 answers

$\int_a^af(x) \, dx$ always $0$?

I was studying integrals and just out of curiosity, Does there exist any 'continuous' functions such that $\int_a^af(x) \, dx$ ($a$ is any number) equals a value other than $0$? Since continuous functions are Riemann integrable, so I think it…
zxcvber
  • 2,081
7
votes
4 answers

Calculating the integral $\int_{-\infty}^{\infty} e^{-x^2}\sin^{2}(2016x)\,dx$

I want to calculate the improper integral $$\int_{-\infty}^{\infty} e^{-x^2}\sin^{2}(2016x)\,dx$$ but I don't know how to change the variable. Please guide me.
math
  • 421
1 2 3
…
99 100