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INTEGRAL-1-0-ln-2-x-2-e-1-x-dx-2-sin-x-cos-x-sin-4-x-cos-4-x-dx-

Question Number 134636 by bobhans last updated on 06/Mar/21 $$\mathcal{INTEGRAL} \\ $$$$\left(\mathrm{1}\right)\int_{\mathrm{0}} ^{\:\mathrm{ln}\:\mathrm{2}} \:{x}^{−\mathrm{2}} .{e}^{−\frac{\mathrm{1}}{{x}}} \:{dx}\:=? \\ $$$$\left(\mathrm{2}\right)\:\int\:\frac{\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}+\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}\:\mathrm{dx}\:=? \\ $$ Answered by benjo_mathlover…

Let-A-a-b-c-d-Find-a-condition-on-a-b-c-d-so-that-A-n-1-A-n-nA-n-N-A-n-1-A-n-nA-A-n-A-n-1-n-1-A-A-n-1-A-n-2-n-2-A-A-n-2-A-n-3-n-3-A-A-4-A-3-3A-A-3-A-2-2A

Question Number 3566 by Yozzii last updated on 15/Dec/15 $${Let}\:{A}=\begin{pmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{pmatrix}.\:{Find}\:{a}\:{condition}\:{on} \\ $$$${a},{b},{c},{d}\:{so}\:{that}\:{A}^{{n}+\mathrm{1}} −{A}^{{n}} ={nA},\:{n}\in\mathbb{N}. \\ $$$$ \\ $$$${A}^{{n}+\mathrm{1}} −{A}^{{n}} ={nA} \\ $$$${A}^{{n}} −{A}^{{n}−\mathrm{1}} =\left({n}−\mathrm{1}\right){A} \\…

Define-the-sequence-a-n-by-the-recurrence-equation-a-n-1-pa-n-qa-n-1-n-1-where-p-q-C-0-and-a-0-a-1-C-Find-a-n-in-terms-of-n-

Question Number 3565 by Yozzii last updated on 15/Dec/15 $${Define}\:{the}\:{sequence}\:\left\{{a}_{{n}} \right\}\:{by}\:{the} \\ $$$${recurrence}\:{equation}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}_{{n}+\mathrm{1}} ={pa}_{{n}} +{qa}_{{n}−\mathrm{1}} \:\:\left({n}\geqslant\mathrm{1}\right) \\ $$$${where}\:{p},{q}\in\mathbb{C}−\left\{\mathrm{0}\right\}\:{and}\: \\ $$$${a}_{\mathrm{0}} =\alpha\:,\:{a}_{\mathrm{1}} =\beta\:\: \\…

Test-for-convergence-1-n-10-2-ln-lnn-nlnn-2-n-2-1-n-lnn-p-two-cases-of-p-to-look-at-3-n-2-1-n-n-lnn-4-n-1-10-n-n-2n-1-5-n-1-

Question Number 3564 by Yozzii last updated on 15/Dec/15 $${Test}\:{for}\:{convergence}: \\ $$$$\left(\mathrm{1}\right)\:\underset{{n}=\mathrm{10}} {\overset{\infty} {\sum}}\frac{\mathrm{2}^{\mathrm{ln}\left(\mathrm{ln}{n}\right)} }{{n}\mathrm{ln}{n}} \\ $$$$\left(\mathrm{2}\right)\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left(\mathrm{ln}{n}\right)^{\mathrm{p}} }\:\left(\mathrm{two}\:\mathrm{cases}\:\mathrm{of}\:\mathrm{p}\:\mathrm{to}\:\mathrm{look}\:\mathrm{at}\right) \\ $$$$\left(\mathrm{3}\right)\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \sqrt{{n}}}{\mathrm{ln}{n}}…

Question-69098

Question Number 69098 by rajesh4661kumar@gmail.com last updated on 20/Sep/19 Commented by Kunal12588 last updated on 20/Sep/19 $${its}\:{really}\:{easy}\:{just}\:{construct}\:{a}\:{line}\:{through} \\ $$$${F}\:{parallel}\:{to}\:{AB}\:\&\:{CD}.\:{and}\:{remember}\:{the} \\ $$$${property}\:{of}\:{alternate}\:{interior}\:{angles}\:{and} \\ $$$${consectutive}\:{interior}\:{angles} \\ $$…