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
4
votes
2 answers

Which answer to follow for the same question?

$$\int \frac {e^x}{(e^x - 1)} dx = \ln|e^x - 1| + C .....A $$ $$\int-\frac {e^x}{(1- e^x)}dx = \ln|1- e^x| + C ......B$$ $$\text{for A: } e^x > 1 \implies e^x > e^0 \implies x>0 $$ $$\text{for B: } e^x < 1 \implies e^x < e^0 \implies…
Sahil
  • 674
4
votes
2 answers

Challenging Integral of $1+\sin^2x+\cdots+\sin^{16}x$

Question Evaluate the integral $$ \int_0^{\frac{\pi}{3}} (1+\sin^2x+\cdots+\sin^{16}x) \ dx$$ Attempt I simplify the GP to $$\frac{1-\sin^{18}x }{ \cos^2x } $$ but at this point, integration seems extremely difficult... This question appeared…
Hugh Entwistle
  • 1,239
4
votes
4 answers

Integral of $x\log(\sin x)$

This is from an old S level paper. I am struggling with part (ii). Any hints?
user85798
  • 1
4
votes
1 answer

Why can't I interchange Integration and Differentiation here?

Consider $f(x,y)=y^3e^{-y^2x}$ and define $F(y) =\int_0^{\infty}f(x,y)dx$ We have that $F'(0)\not = \int_0^{\infty} \frac{\partial f}{\partial y}(x,0)dx$ in the spoiler there is how I got this, in case I made a mistake there We calculate $F'(0)$…
Moritzplatz
  • 1,521
  • 14
  • 28
4
votes
7 answers

Is the area of a line = 1?

I tried to teach my son multiplication using a rectangle. (e.g. 3cm * 4cm = 12cm^2). Now I have 12 little squares. But how do I explain where the "little square" came from? My best guess: If I cut a little square in two, I can get a rectangle 2cm…
Thomas
  • 41
  • 1
  • 1
  • 2
4
votes
1 answer

Using U-Substitution for Integrals

What is the integral of the following: $$\frac{\sin 2x}{1+\sin x}$$ I know that $\sin 2x = 2\sin x \cos x$ and then I substituted $u$ for $\sin x$ but then I got stuck after that. Can you help?
Lizi
  • 61
4
votes
2 answers

Computing $\int\limits_0^{\pi/2} {1 \over 1+8\sin^2(\tan x)}\ dx$

I’m trying to evaluate$$\int\limits_0^{\pi/2}dx\,\frac 1{1+8\sin^2(\tan x)}$$And made the substitution $u=\tan x$. Therefore$$I=\int\limits_0^{\infty}dx\,\frac 1{(1+x^2)(1+8\sin^2x)}$$Through some manipulations, you arrive…
Crescendo
  • 4,089
4
votes
1 answer

How can one check that a definite integral has been evaluated correctly?

In many problems we can check our solution by "solving the problem backwards". E.g. we can plug the solutions of an equation into the equation to get identities, or differentiate the antiderivative of a function we (indefinitely) integrated to get…
Ruslan
  • 6,775
4
votes
2 answers

How to solve $\int(\cos(x)^{\cos(x)+1}\tan(x) (1+\log(\cos(x)))dx$? (2017 MIT Integration Bee Qualifier Problem #20)

I'm working my way through the qualifier, trying to learn some new techniques as I go. I am pretty stymied by this one. Solve: $$\int(\cos(x)^{\cos(x)+1}\tan(x) (1+\log(\cos(x)))dx$$ My steps so far have been to let $u=\cos(x)$ and…
Mitchell Jacky
  • 132
4
votes
1 answer

Integral of Weibull distribution

How would you find the distribution function for the following density functions (Weibull function): $$f_{X}(x) = c\tau x^{\tau−1}e^{− cx^{\tau}} $$ for $0< x < \infty$, $\tau > 0$ and $c>0$.
Sharingan
  • 153
  • 1
  • 1
  • 4
4
votes
2 answers

Advanced integration, how to integrate 1/polynomial ? Thanks

I have been trying to integrate a function with a polynomial as the denominator. i.e, how would I go about integrating $$\frac{1}{ax^2+bx+c}.$$ Any help at all with this would be much appreciated, thanks a lot :) ps The polynomial has NO real roots…
Justin
  • 71
4
votes
1 answer

How to show that $\int_{0}^{1}\left({x^u\over \ln(x)}+{x^v\over 1-x}\right)\mathrm dx=\gamma-H_v+\ln(u+1)?$

It is well-known $$\int_{0}^{1}\left({1\over \ln(x)}+{1\over 1-x}\right)\mathrm dx=\gamma\tag1$$ Messing around with $(1)$ using wolfram integrator, we have $$\int_{0}^{1}\left({x^u\over \ln(x)}+{x^v\over 1-x}\right)\mathrm…
gymbvghjkgkjkhgfkl
  • 8,090
4
votes
1 answer

Evaluate $\int_{0}^{\infty}e^{-{1\over x}}\sin\left({1\over x}\right)\ln(x){\mathrm dx\over x^{4n+1}}$

$$\int_{0}^{\infty}e^{-1\over x}\sin\left({1\over x}\right)\ln(x){\mathrm dx\over x^{4n+1}}=(-1)^{n+1}\pi F(n)\tag1$$ $n\ge1$ $F(1)={3\over 8}$ $F(2)={315\over 4}$ $F(3)=155925$ I can't seem to figure the pattern for $F(n)$ How can we evaluate the…
gymbvghjkgkjkhgfkl
  • 8,090
4
votes
3 answers

Changing limits of integration on a difficult integral

I have the following integral: $$\int_0^1 \int_0^{y^5} 7y^8 e^{xy^2} \, dx \, dy$$ I tried drawing a picture and finding new limits of integration but the integral was still difficult to solve which means my new limits were wrong. Help is…
MilTom
  • 365
4
votes
1 answer

Find $\int{\frac{1}{x^4\arctan(x)+x^3+x^2\arctan(x)+x}}dx$

I've been struggling with this one for a while, but couldn't do it. $$\int{\frac{1}{x^4\arctan(x)+x^3+x^2\arctan(x)+x}}dx$$ What I tried was factoring like so: $$I = \int{\frac{1}{(x^2+1)(x+x^2\arctan(x))}}dx$$then substitute $u=\arctan(x)$, which…
NotADeveloper
  • 731
  • 1
  • 4
  • 10
1 2 3
…
99 100