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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Help with finding integral

I've been trying for an embarrassingly long time to figure this one out. It looks like it should crack under integration by parts and integration by substitution, but I am having trouble with it. Any pointers? $$\int {\exp(a \sqrt{x^2 + b} ) \over…
Mark Borgerding
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Solve $a$ and $b$ for centre of mass in $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Given ellipse: $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$ What length do $a$ and $b$ have to be so the centre of mass is $S(4;2)$? I've tried steps to solve the equation to $$y=b\sqrt{1-\frac{x^2}{a^2}}$$ and integrate…
user1211030
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Calculate the length of curve

The curve is the intersection of: $$4x=(y+z)^2$$ $$4x^2+3y^2=3z^2$$ And the interval of curve length is from $O(0,0,0)$ to $M(x,y,z)$ The answer is $\sqrt2 z$ My substitution is $u=y+z$ and $v=z-y$, then I put them into these two equations…
Zhen Zhang
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Calculate integration with Euler integration

Calculate: $$\int^{\pi /2}_{0}\tan ^{\alpha} x dx$$ with $| \alpha|<1$ Answer: $\pi/(2 \cos \frac{\alpha \pi}{2})$ In $B$ function or integral, we have $B(x,y)=\int^{1}_{0}t^{x-1}(1-t)^{y-1}dt$ which is defined on interval $(0,1)$, but the problem…
Zhen Zhang
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Line Integral of a circle

I have this question from "Div,Grad,Curl and all that" by Schey I have: $$F =e_{\theta}/r$$ Find the line integral F t from the point $P_{1}(0,-1,0)$ to point $P_{2}(0,1,0)$ over two difference paths:$C_{R}$, the right hand side of the circle of…
user -1
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Exponential integration rule

Given the function $e^ae^{-ba}$ The indefinite integral is $\int e^ae^{-ba} \mathrm da = \int e^{a-ba} \mathrm da$ is $\frac{e^ae^{-ba}}{1-b}$ I get that $\int e^u = e^u\frac{\mathrm du}{\mathrm da}$. I cannot seem to understand how the $(1-b)$ term…
strimp099
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Notation of an infinite integral

I'm confused regarding the notation of the integral I've underlined in green. Does this mean that the integral over any range is $\infty$?
usainlightning
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Need help finding this Antiderivative

I have no idea how to do that, help please I was trying a change of variables in integration but nothing $$ \int \frac{x3^{x-a}}{3^{3x-a} + 3^{2x+1} + 3^{x+a+1} + 3^{2a}}\, dx $$
ZellAllon
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Please someone help with this nearly impossible integral

$$ \int \frac{4x^5 -1} {(x^5 + x +1)^2} dx $$ So this is the integral and I have been stuck on it for ages without getting anywhere at all. Nothing I tried has gotten me anywhere so I'm basically stuck on nowhere with this. I would really love for…
user135839
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Integration by parts or substitution for $\int x \cos (x^2) dx$

With $$\int x \cos (x^2) dx$$ I can use substitution to solve this integral. $$u=x^2, dx =\frac{du}{2x}$$ $$\int x \cos (x^2) dx = \int x \cos (x^2) \frac{du}{2x} = \frac{1}{2} \int \cos(u) = \frac{1}{2}\sin(u)=\frac{1}{2}sin(x^2)+C$$ But can't I…
Chris
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How to integrate $x\sin(1+2x)$

Can someone be kind enough to show me the steps to integrate this, I'm sure it's by parts but how do I split up the sin part? $$x\sin(1+2x)$$
Helpthanks
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Evaluate $\int^{\pi/3}_0 \sin x \ln (\sec x)\text{dx}$

$\int^{\pi/3} _0 \sin x \ln (\sec x)\text{dx}$ So far I have: $u=\ln(\sec x), v'=\sin x$ $u'=\cos x, v=-\cos x$ $[-\cos x \ln (\sec x)-\int-\cos^2 x dx]^{\pi/3}_0$ Is this right so far?
Jim
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Integration by substitution help please

$$ \int_0^{\pi/2} \frac{\sin\theta}{1+\cos\theta} \, d\theta $$ My working thus far: $$u=1+\cos\theta$$ $$\text{d}u=-\sin\theta \ \text{d} \theta$$ Substituting limits in and obtaining them in terms of u: $$\int^1_2 \frac{\sin\theta}{u} \cdot…
Jim
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Integration by substitution - $\int (x+1)\sqrt{5x-6} \text{dx}$

$\int (x+1)\sqrt{5x-6} \text{dx}$ My working so far: $u=5x-6$ $du=5dx$ $\int\dfrac{x+1}{5}\sqrt u \ \text{du}$ Substituting x $\int \dfrac{(u+6)/5)+(5/5)}{5}\cdot\sqrt u \ \text{du}$ $\int (u+11)\sqrt u \ \text{du}$ I've got the same powers right…
Jim
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Integration by substitution - $\int x^2 \sqrt{x-2} \text{dx}$

$$\int x^2\sqrt{x-2} \, dx,u=x-2$$ Using the given substitution $u=x-2$ $\text{du}=\text{dx}$ Attempting to express integral in terms of u... $\int u^2+4x-4\cdot \sqrt{u} \ \text{du}$ This is where I'm stuck - where have I gone wrong?
Jim
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