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
3
votes
1 answer

Tedious Trig Integral: $\int\limits_{0}^{\pi/2}{\frac{\theta (1+\sin^2\theta)\cos\theta}{(1+3\sin^2\theta)(3+\sin^2\theta)}d\theta}$

$$\int\limits_{0}^{\pi/2}{\frac{\theta \left( 1+\sin^{2}\theta \right)\cos \theta }{\left( 1+3\sin^{2}\theta \right)\left( 3+\sin^{2}\theta \right)}d\theta }$$ Attempt: I tried making the substitution $u = \sin(\theta)$ so that the integral could…
Why Do You Care
  • 335
3
votes
6 answers

How can I integrate $\int \arctan(\sec x + \tan x) dx$

I got this problem in my homework exercise: $$\int \arctan(\sec x + \tan x) dx$$ I simplified it to $$\int \arctan\left(\dfrac{1+\sin x}{\cos x}\right) dx$$ $$=\int \arctan\left(\sqrt{\dfrac{1+\sin x}{1-\sin x}}\right) dx$$ now, I tried putting…
jonsno
  • 7,521
3
votes
5 answers

Find $\int \frac{x^2}{(x \cos x - \sin x)(x \sin x + \cos x)}dx $

Find $$\int \frac{x^2}{(x \cos x - \sin x)(x \sin x + \cos x)}dx $$ Any hints please? Could'nt think of any approach till now...
user220382
3
votes
1 answer

How to solve $ \int\sqrt{e^x}$ - different approaches seem to yield different results

Approaching the following integral in different ways appears to yield different results: $\int \sqrt{e^x} dx$ Simplifying $\sqrt{e^x}$ to $(e^x)^\frac{1}{2}$ to $e^\frac{x}{2}$. Now, integrating simply takes one over the inner derivative as the…
Marco
  • 189
  • 7
3
votes
1 answer

Integral of $\frac{1}{|t-s|^p}$; when is it finite?

Let $p > 1$ be a number. How can I tell for which values of $p$ the integral $$\int_0^T \int_0^T \frac{1}{|t-s|^p}\;dtds$$ is finite? Here $T$ is a positive and finite number. The singularity is when $t=s$ where we have the problem. but I don't know…
RequiredButNeverShown
  • 33
3
votes
3 answers

Evaluation of $\int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt+\int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt$

$$\int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt+\int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt$$ $\bf{My\; Try::}$ Let $$f(x) = \int_{\frac{1}{e}}^{\tan x}\frac{t}{1+t^2}dt+\int_{\frac{1}{e}}^{\cot x}\frac{1}{t(1+t^2)}dt$$ Put $\displaystyle x =…
juantheron
  • 53,015
3
votes
2 answers

Compute $\int \sqrt{1+4x^2} \, dx$ with Euler substitution

In this post: Computing $\int \sqrt{1+4x^2} \, dx$ someone mentioned Euler substitution to compute the following integral: $$\int \sqrt{1+4x^2} \, dx$$ I tried to follow this advice and got very nice result, namely I substituted $\sqrt{1+4x^2}=t-2x$…
Paweł Orliński
  • 626
3
votes
0 answers

integral of sine cubed times cosine cubed

For the $\int \sin^3 x\cos^3 x \,dx$ I got $\frac 16(cos x)^6 - \frac14cos^4 x + C$. This seemed to work but my book had $\frac 14\sin^4 x - \frac 16 sin^6 x + C$. This is similar to what I got but instead of cosine they had sine and they had the…
alex
  • 31
3
votes
1 answer

Solving integral without partial fraction

I would like to expand my pool of integral solving skills and thus try to solve older problems again, however with a different method I had used back then, when I encountered them first. For this problem I omitted the lower and upper bound and just…
Imago
  • 2,132
3
votes
2 answers

Integration Boundaries for Marginal Distribution

If the joint distribution of $x$ and $y$ as follow $$f(x,y)=\left\{ \begin{align} & \frac{{{x}^{2}}+y}{4}\,\,\,\,,\,\,\,\,0
SwabianOrtolan
  • 129
3
votes
3 answers

Prove $\int_{0}^{1}{1\over 1+x^2}\ln\left({1+x\over 1-x}\right)=K$

Showing $$I=\int_{0}^{1}{1\over 1+x^2}\ln\left({1+x\over 1-x}\right)=K\tag1$$ $K=0.9159655...$ ;Catalan's constant. Recall $$\ln\left({1+x\over 1-x}\right)=2\sum_{n=0}^{\infty}{x^{2n+1}\over 2n+1}\tag2$$ Sub (2) into (1)$\rightarrow…
gymbvghjkgkjkhgfkl
  • 8,090
3
votes
2 answers

Why is this incorrect $\int_{0}^{1}{\ln(x)\over (1+x)^3}dx=-\sum_{n=0}^{\infty}{(-1)^n(n+2)\over 2(1+n)}$

$$I=\int_{0}^{1}{\ln(x)\over (1+x)^3}dx$$ Recall $${1\over (1+x)^3}=\sum_{n=0}^{\infty}{(-1)^n(n+1)(n+2)\over 2}x^n$$ $$\int_{0}^{1}x^n\ln(x)dx=-{1\over (n+1)^2}$$ Substitute in $$I=\sum_{0}^{\infty}{(-1)^n(n+1)(n+2)\over…
gymbvghjkgkjkhgfkl
  • 8,090
3
votes
2 answers

Evaluate this $\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}dx$

$$\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}dx$$ $$\int_{0}^{\infty}{x^s-1 \over x-1}\cdot{1-e^{-ax} \over 1-e^{ax}}\cdot{e^{ax}\over e^{ax}}dx$$ $$\int_{0}^{\infty}-{x^s-1 \over x-1}\cdot{e^{-ax}}dx$$ Suppose…
gymbvghjkgkjkhgfkl
  • 8,090
3
votes
1 answer

How to integrate $\int \arctan(e^{-\pi y/b}) \, dy$?

How to integrate $\int \arctan(e^{-\pi y/b}) \, dy$
Am.Jafari
  • 31
3
votes
2 answers

Integrate $\int e^x\frac{1+\sin x}{1+\cos x}dx$

Integrate $$\int e^x\cdot\frac{1+\sin x}{1+\cos x}\,dx$$ My try; First step: I let $$\frac{1+\sin x}{1+\cos x} = u$$ $$e^x = v$$ and then I applied integration by parts: $$\frac{1+\sin x}{1+\cos x}=u \implies du=\frac{\sin x+\cos x+1}{(1+\cos…
Jale'de jaled
  • 780
1 2 3
…
99 100