All aspects of integration, including the definition of the integral and computing different types of integrals. For questions solely about the properties of integrals, don't use this tag alone! Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another ...

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7
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2answers
182 views
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Integrate $\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$

Integrate $$I=\int\sqrt\frac{\sin(x-a)}{\sin(x+a)}dx$$ Let $$\begin{align}u^2=\frac{\sin(x-a)}{\sin(x+a)}\implies ...
1
vote
0answers
41 views
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If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)$, then the curve $P(x,y)=\alpha$ cuts $y=f^{-1}x$ at?

If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)\forall x,y\in\mathbb R$, then the curve $$y^2+\int_0^x(\sin t+a^2t^3+bt)dt=\alpha,\alpha\in\mathbb R^+$$ cuts $y=f^{-1}x$ ...
1
vote
1answer
97 views
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Closed form of $\int_{a}^{b}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx$ for some $a<b$

In this question I asked to prove that $$\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}.$$ If we take a look at the plot of the integrand, then we could see some symmetry-property. ...
8
votes
2answers
188 views
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Calculation of $\int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$

Calculate $$ \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2 \cos^2 x}}dx$$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{4}}\tan^{-1}\sqrt{\frac{\cos 2x }{2\cos^2 ...
5
votes
4answers
104 views
+50

Software, techniques and tricks of experimental mathematics to conjecture possible closed forms

It often happens that people conjecture possible closed forms of integrals, series, and so on starting from a numerical value calculated to very high precision. What are the techniques, tricks, ...