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
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i found two solutions for the integral $\int \frac{x}{1-x}\,dx$, but one is wrong. Why?

I have math problem $\int \frac{x}{1-x}\,dx$ and found two solutions for it: 1.$\int \frac{x}{1-x}\,dx=\int \frac{x-1+1}{1-x}\,dx=\int \frac{-1+x}{1-x}+\frac{1}{1-x}\,dx=\int…
urshuk
  • 81
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Primitive of $u'' /u$

There is a primitive for $$f'(x)/f(x)$$ which is $\ln(f(x))$ but is there any known primitive for $f''(x)/f(x)$ ?
BenG73
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Difficulty obtaining anti-derivative

This was evidently a question in a recent Calculus 1 exam paper. $$ \int_{0}^{0.5} \frac{(16x^2-8x+1)\exp(\cos(\pi x)}{\exp(\cos(\pi x)+ \exp(\sin(\pi x)}dx $$ I have several attempt to find an analytical expression for the anti-derivative if the…
Callie12
  • 581
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Calculate the area enclosed by a curve with equation $\theta=f(r)$

I need to calculate the area of an object that is limited by this line: $$\phi=r\!\cdot\!\arctan(r)\;,\quad\phi\in\left[0,\frac{\pi}{\sqrt3}\right]$$ I am using this formula: $$S=\frac12\int_{r_1}^{r_2}r^2\phi'\mathrm dr$$ When I plugged in values I…
ALiCe P.
  • 315
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What is the primitive of $\dfrac{1}{\sin x+1}$?

I hope you are well I try to find the infinite primitive of $\dfrac{1}{\sin x+1}$ without using the méthode of substituting by $\tan \frac x2$. So I multiply and divide by $\sin x - 1$ and simplify until having The result: $\tan x -\dfrac{1}{\cos…
Alia
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Find value of a line integral using Stokes' theorem

Given a vector field $\vec A = y\vec i + z\vec j + x\vec k$ Use Stokes' formula to calculate the line integral $\oint\limits_C {\vec Ad\vec r} $ in here, $C$ is a circle ${x^2} + {y^2} + {z^2} = {a^2}$ intersects $x + y + z = 0$
user38457
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Indefinite integral $\int x\cot^2x\,dx$

Approach that I have tried: $u = x$ $\to$ $u' = 1$ $v'= \cot x\to v = \ln(\sin x)$ $$\int x\cot^2x\,dx=x\ln(\sin x)-\int\ln(\sin x)\,dx$$ And from there, I have a problem, because solving for integral of ln(sinx) gives complicated results with I…
user4t48u
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Evaluate $\int\;\frac {x^m\ln(x)}{\cos(x)}dx$

Question: How to Evaluate Integral $$\int\;\frac {x^m\ln(x)}{\cos(x)^2}\mathrm{d}x$$ My suggestion $$F(x)=\int\;\frac {x^m\ln(x)}{\cos(x)^2}\mathrm{d}x$$ integration by part let $$\mathrm{d}u = \frac{1}{\cos(x)^2} \; \; v = x^m\ln(x)\\ \implies…
Moustapha_M_I
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help regarding an integral

I have the following integral: $$ I=\int_{V=0}^{1} \min (1,(u-1)^{n-1}(u-V)^{n-1}) dV, $$ where u is a constant which can take values between 1 and 2 and $V$ is variable whose range is (0,1). putting $u-V=t,$ we get : $$ I= \int_{t=u-1}^{u} \min…
AgnostMystic
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Are there any errors in my calculation of $I_{0}(t)=\lim_{w\to w_0}I(t),$ and my proof that $I(t)$ is bounded and $I_0(t)$ is not?

Here is my problem. I reproduce the two questions below. I'm not satisfied at all with how I solved this. Are there any errors? $$\begin{split} I(t)& = \frac{2A}{w_{0}^2-w^2}\sin\frac{(w_{0}-w)t}{2}\sin\frac{(w_{0}+w)t}{2}\\ I_{0}(t)&=\lim_{w\to…
Alexander Aseme
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Reduction Integration Question ($\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$)

What integration by parts things do I use for a reduction formula for $\int_0^1{\frac{x^n}{\sqrt{x+1}} dx}$? I have tried many different ways of obtaining it without success.
Reese
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Integrals by substitution: general case?

I am trying to understand a general case for the substitution rule. My real case is this: $$\int{}(\sqrt{9-t^2})(-2t)dt$$ Making a generality. If I have an integral in the form of: $$\int{ab}$$ Where a need to be treated by the SUBSTITUTION RULE…
Haizum Skallah
  • 653
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Conditions for Equality in Fatou's Lemma other than Hypotheses for Classical Convergence Theorems?

Let $(X, \mathcal{M}, \mu)$ be a measure space, and let $\{f_n\}_{n=1}^{\infty}$ be a sequence of nonnegative extended real-valued measurable functions defined on $X$. Suppose also that $\lim_{n\to\infty}f_n=f$. Are there any conditions which can…
Timothy
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Integration problem with substitution rules

I have to make the integration with the substitution rule: $$\int {(x^2-1})^2(2x)dx$$ $$u=x^2-1$$ $$\int {\sqrt{u}}du=\frac{2u^{3/2}}{3}$$ $$\frac{2(x^2+1)^{3/2}}{3}(2x)$$ My question is: Do I need to integrate de las part of the expression: the 2x?…
Haizum Skallah
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Calculate the double integral

Compute $$\int^{\infty}_{-\infty}{\int^{\infty}_{0}{xe^{-x^2e^y}}}dydx$$ Since x is just a constant in the inner integral I took it out and was only left with $\int^{\infty}_{0}{}e^{-x^2e^y}dy.$ This is where I'm stuck. I cannot figure out this…
Techlover
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