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.

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Integral $\lim_ {t\to0}\frac{1}{t}\int_t^{2t}\frac{\ln(1+x)}{{ {\sin x}}}\ dx$

How to integrate $$\lim_ {t\to0}\frac{1}{t}\int_t^{2t}\frac{\ln(1+x)}{{ {\sin x}}}\ dx\ ?$$Is it okay to use L'Hospital or this method can't be used if inside the integral its $\frac{0}{0} $. I asked this question before but I wasn't attentive…
Lola
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Integration Using Trig Sub and Partial Fractions

$\displaystyle\int\frac{1}{x^4 + x^2 + 1}\mathrm{d}x$ using the trig substitution. My attempt: I got $\left(x^2 + \frac{1}{2}\right)^2+\frac{3}{4}$ and did $x= \frac{1}{\sqrt2}\tan \theta$ but did not know how to proceed because I got…
user245640
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Primitive of $\sin$ function (Weierstrass)

How do I find the primitive of the following function: $$f(x)=\frac{{\sin x}}{{1+\sin x}}.$$ I solved this with Weierstrass and I found it's: $$2\int(1+t)^{-2}dt$$ where $t=\tan(x/2)$ and if I solve it this would mean the integral is:…
Lola
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Find $\int x\times\sqrt{8-x^2} \,dx$

$$\int x\times\sqrt{8-x^2} \,dx = \,?$$ I got to this: $$\int\sqrt{8x^2-x^4} \,dx$$ or: $$\int\frac{8x-x^3}{\sqrt{8+x^2}}\, dx$$ I don't know how to integrate neither. If possible, no $\sin$ \ $\cos$ \ etc.
yuvalz
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How to integrate $\int 2xe^{x^2-y^2}\cos(2xy)- 2ye^{x^2-y^2}\sin(2xy) \ \mathrm dy$?

Find the value of: $$\int 2xe^{x^2-y^2}\cos(2xy)- 2ye^{x^2-y^2}\sin(2xy) \ \mathrm dy$$ i have tried using integration by parts but it doesn't seem to work. How would I go about this integral ?
april analysis
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Evaluate the following integral with variable upper limit

Evaluate the integral $$J(d,x)=\int_0^x\frac{1}{\sqrt{d^2-\sin(r)-\cos(r)}}dr;\space d^2>\sqrt2$$ I am unsure where to begin with this. Any help appreciated.
Kirin171
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Does $\int _0^1 \cfrac{x}{x+\cfrac{x^{2}}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$ exist?

$$\int_0^1 \cfrac x {x+\cfrac{x^2}{x^2+\cfrac{x^3}{x^3+\cdots}}} \, dx$$ Does this integral exist? Any hint will be appreciated.
z3wood
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Is it proper to integrate an expression containing units?

Is it proper to integrate an expression such as $\displaystyle \int \frac 1 {x~\mathrm{J}} \, dx$, where $x$ is in the physical unit of Joules (J)? The result is $\ln \dfrac x J + \text{constant}$. However, I don't know how to take the natural log…
Mark Ciotola
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How to integrate $\int e^{-t}f(t)dt$ for arbitrary $f(t)$

Say you have the integral $$g(t)=\int e^{-t}f(t)dt$$ Is it possible to rewrite this in some way such that we can find $g(t)$ for any $f$, so long as $f$ is itself tractable? E.g. is there a way to rewrite it as $g(t)= A(t)+ B(t)\int f(t)dt$ for…
user56834
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Fresnel Integral

Does someone know how they go from the left side to the right side? $$\left(\int_{0}^{R}e^{-x^2/2}\,dx\right)^2 = \iint_{[0,R]^2} e^{-(x^2+y^2)/2}\,dx\,dy $$ Thanks.
Ayoub Rossi
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Evaluate $\lim _{n\to \infty }n\int _1^2 \frac{dx}{x^2(1+x^n)}$

Evaluate $$\lim _{n\to \infty }n\int _1^2 \frac{dx}{x^2(1+x^n)}$$ without Taylor expansion. I tried rewriting as $$\lim_{n\to \infty} n\int _1^2 \frac{1 + x^n - x^n}{x^2(1+x^n)}dx = \lim_{n\to \infty} n\int _1^2 \frac{dx}{x^2} - \lim_{n\to \infty}…
Liviu
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Which value of $a$ maximizes $\int_{a-1}^{a+1}\frac{1}{1+x^{8}}dx$?

I am not being able to understand the graphical method of solving this, any simple explanation will be appreciated. A non-graphical calculation will be very helpful too. Thank you so much in advance!
Idkwoman
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Integration by first principle?

Is there a "formal" counterpart (or equivalent) to the process of differentiation by first principle for computing integral?
user12345
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integration and limit of a function

Let $\,\,f:\mathbb{R}\rightarrow\mathbb{R}\,\,$ be a continuous function such that $ \ \int_{0}^{\infty}{f(x)} dx $ exists. Which of the following statements are always true ? 1.if $\lim_{n\rightarrow\infty}f(x)$ exists, then…
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