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Oct
13
comment Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
@Danxe, From my answer, $$I=\int_0^\pi(\pi-x)f(\sin x)dx=\pi\int_0^\pi f(\sin x)dx-\int_0^\pi xf(\sin x)dx$$ $$\implies I=\pi\int_0^\pi f(\sin x)dx-I$$
Oct
13
comment Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
@Danxe, Setting $a+b-x=u,-dx=du$ $$\int_a^bf(x)dx=\int_b^af(a+b-u)(-du)=-\int_a^bf(a+b-u)du$$ Now, $$\int_a^bf(a+b-u)du=\int_a^bf(a+b-x)dx$$ Finally, $$\int_c^df(x)dx=-\int_d^cf(x)dx$$
Oct
13
comment Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
@Danxe, Set $a+b-x=u$ in the left hand side
Oct
13
answered Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$
Oct
13
answered Substitution method for solving recurrences piece wise function
Oct
13
comment Trigonometry equation with arctan
@XMLParsing, How about this?
Oct
13
answered Trigonometry equation with arctan
Oct
13
answered Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
Oct
13
answered How to solve $\frac{15!}{(x-1)!(16-x)!}=\frac{15!}{(2x+1)!(14-2x)!}$ for $x$?
Oct
12
answered $\lim_{x\to\infty} \frac{x}{\sqrt{9x^2 +x} -3}$
Oct
12
comment How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator?
@user183782, How about this?
Oct
12
answered How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator?
Oct
12
comment How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator?
Please rewrite the first line to make it readable
Oct
12
comment Calculating $3/10$ in $\mathbb{Z}_{13}$
@user160292, $$10^{-1}\equiv10^{11}\equiv-3^{11}\equiv4$$ $$\implies3\cdot10^{-1}\equiv3\cdot4\pmod{13}$$
Oct
12
comment Calculating $3/10$ in $\mathbb{Z}_{13}$
@Alan, I think one should be able to reach at the answer if one understands the method. Spoon feeding may not be recommended
Oct
12
comment Calculating $3/10$ in $\mathbb{Z}_{13}$
@user160292, I think yours is trial & error. Checking the same for larger modulo can prohibitively complex
Oct
12
answered Calculating $3/10$ in $\mathbb{Z}_{13}$
Oct
12
comment How to prove $\sum_{n=0}^{\infty} \frac {(2n+1)!} {2^{3n} \; (n!)^2} = 2\sqrt{2} \;$?
Good Job. One quick question : how have you derived the Right Side of the Identity from the left?
Oct
12
answered How to simplify $\sqrt[3]{29\sqrt{2}-45}-\sqrt[3]{29\sqrt{2}+45}$
Oct
12
answered Prove that $n|(n+1)^n - 1$