# Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

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### area under the curve $1/(2x)$

How can I find the area under the curve of $y = 1 / (2x)$ in the intervals $0 <x, y <1?$
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### Calculate $\int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$

I'm trying to evaluate $$\int_0^{2\pi} \ln(2-2\cos(t)) \ln(2-2\cos(t+\theta)) dt$$ but I'm not sure the best way to proceed. I've been trying to factor the inner terms in to rational functions of ...
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### Find the volume between surfaces

I'm trying to find the volume between the surface $x^2+y^2+z=1$ and $z=x^2+(y-1)^2$ but nothing works for me. I made the plot and it looks like this: How could you start? Any recommendation?
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### How to find a volume of an object enclosed with planes without any projection?

How to find a volume of an object enclosed with planes: $$x^2+z^2=4,$$ $$x+y=2,$$ $$x+y=-2,$$ $$x-y=2,$$ $$x-y=-2$$ without any projection? When I project this object I know it is a truncated ...
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### Convergence $\int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$

I was trying to solve the following integral: $$\int_{-1}^1 \sqrt{1-\frac{x}{(1-x^2)^2}}$$ But when I plugged it in to any online calculator, It said it couldn't find the integral and that it might ...
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### Comparing summation and integration for non monotonic function

$$P=\sum_{r=3n}^{4n-1} \frac{r^2+13n^2-7rn}{n^3}$$. $$Q=\sum_{r=3n+1}^{4n} \frac{r^2+13n^2-7rn}{n^3}$$. $$I=\int_{3}^{4} (x^2-7x+13) dx = \frac{5}{6}$$ Compare the values of $P,Q,I$ I know ...
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### Need help to solve this integral. Don't know how to do [closed]

I need help to evaluate$$\int_1^2\frac{x^3+4}{x^2+2x}dx$$
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### How can I calculate this integral considering the point where denominator is zero?

$$I=\int_{0}^{3}\frac{1}{(y-1)^\frac{2}{3}}dy$$ If I substitute $u=y-1$ and split this integral up and calculate it for $0≤y<1$ and $1<y≤3$ then I get the anwer $I=3(2^\frac{1}{3}+1)$. But then ...
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### Integration limits at points where denominator is zero

If the domintator of some function $f$ from $[a,b]$ to $ℝ^2$, be equal to zero at some point $c∈[a,b]$, does it necessarily imply that the function is not integrable on $[a,b]$? If yes, how should the ...
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Evaluate $$I=\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$$ We have $$\frac{\ln(1-x)}{x}=-\sum_{k=1}^{\infty}\frac{x^{k-1}}{k}$$ Hence $$I=-\sum_{k=1}^{\infty}\left(\frac{1}{k}\int_{0}^{1}x^{k-1}\... 3answers 220 views ### How to solve this integral with transformation to polar coordinates? How do I determine new limits when transforming to polar coordinates. I have this example, and I don't know how to solve it correctly.$$ \iint_D \frac{\ln\left(x^2+y^2\right)}{x^2+y^2}\,dx\,dy $$... 2answers 115 views ### Ahmed integral revisited \int_0^1 \frac{\tan ^{-1}\left(\sqrt{x^2+4}\right)}{\left(x^2+2\right) \sqrt{x^2+4}} \, dx How can we prove (it is numerically verified already):$$\int_0^1 \frac{\tan ^{-1}\left(\sqrt{x^2+4}\right)}{\left(x^2+2\right) \sqrt{x^2+4}} dx=-\frac{5 \pi ^2}{16}-\frac{1}{4} \tan ^{-1}\left(\sqrt{...
There is a second order integral which looks like a fourier transform as shown below.  \int_{-a/2}^{a/2} \int_{-a}^{a} \left(1-\frac{|x|}{a}\right) e^{jkx \sin\theta \cos\phi} e^{jky \sin\...