Question and Answers Forum

AllQuestion and Answers: Page 1368

Question Number 76974 Answers: 1 Comments: 0

Calculate the side of an equilateral triangle whose vertices are situated on three parallel coplanar lines, knowing that a and b are the distances of the parallel line to the others.

$${Calculate}\:{the}\:{side}\:{of}\:{an}\:{equilateral} \\ $$$${triangle}\:{whose}\:{vertices}\:{are}\:{situated} \\ $$$${on}\:{three}\:{parallel}\:{coplanar}\:{lines}, \\ $$$${knowing}\:{that}\:\boldsymbol{{a}}\:{and}\:\boldsymbol{{b}}\:{are}\:{the}\:{distances} \\ $$$${of}\:{the}\:{parallel}\:{line}\:{to}\:{the}\:{others}. \\ $$

Question Number 76973 Answers: 2 Comments: 0

In a ABC triangle the side a=6 and c^2 −b^2 =66. Calculate the projections of sides b and c on a.

$${In}\:{a}\:{ABC}\:{triangle}\:{the}\:{side}\:\boldsymbol{{a}}=\mathrm{6}\:{and} \\ $$$$\boldsymbol{{c}}^{\mathrm{2}} −\boldsymbol{{b}}^{\mathrm{2}} =\mathrm{66}.\:{Calculate}\:{the}\:{projections} \\ $$$${of}\:{sides}\:\boldsymbol{{b}}\:{and}\:\boldsymbol{{c}}\:{on}\:\boldsymbol{{a}}. \\ $$

Question Number 76967 Answers: 0 Comments: 0

∫_0 ^∞ ((cos((√x)))/(e^(2π(√x)) −1))dx+Σ_(n=0) ^∞ (n/e^(n ) )=?

$$\int_{\mathrm{0}} ^{\infty} \frac{{cos}\left(\sqrt{{x}}\right)}{{e}^{\mathrm{2}\pi\sqrt{{x}}} −\mathrm{1}}{dx}+\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{{n}}{{e}^{{n}\:} }=? \\ $$

Question Number 76965 Answers: 1 Comments: 0

Evaluate I_(ab) =∫sin axcos bxdx if a≠b and use it to ∫_0 ^n sin 3xcos 2xdx=((3−(√3))/5)

$${Evaluate} \\ $$$${I}_{{ab}} =\int\mathrm{sin}\:{ax}\mathrm{cos}\:{bxdx} \\ $$$${if}\:{a}\neq{b}\:{and}\:{use}\:{it}\:{to} \\ $$$$\int_{\mathrm{0}} ^{{n}} \mathrm{sin}\:\mathrm{3}{x}\mathrm{cos}\:\mathrm{2}{xdx}=\frac{\mathrm{3}−\sqrt{\mathrm{3}}}{\mathrm{5}} \\ $$

Question Number 76964 Answers: 1 Comments: 0

solve Differential equation x^2 (d^2 y/dx^2 )−4x(dy/dx)+3y=x^3 +2x+5

$${solve}\:{Differential}\:\:{equation} \\ $$$${x}^{\mathrm{2}} \frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }−\mathrm{4}{x}\frac{{dy}}{{dx}}+\mathrm{3}{y}={x}^{\mathrm{3}} +\mathrm{2}{x}+\mathrm{5} \\ $$

Question Number 76963 Answers: 1 Comments: 0

∫cos 2θ ln (((cos θ+sin θ)/(cos θ−sin θ)))

$$\int\mathrm{cos}\:\mathrm{2}\theta\:\mathrm{ln}\:\left(\frac{\mathrm{cos}\:\theta+\mathrm{sin}\:\theta}{\mathrm{cos}\:\theta−\mathrm{sin}\:\theta}\right) \\ $$

Question Number 76958 Answers: 1 Comments: 1

∫ln e^y dy

$$\int{ln}\:{e}^{{y}} {dy} \\ $$

Question Number 76956 Answers: 1 Comments: 0

∫_( 0) ^( a) (dx/(√((a+sinx)(a+cosx)))) =? [a∈R^+ ,test for: a=1]

$$\underset{\:\:\mathrm{0}} {\overset{\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{a}}} {\int}}\:\:\:\frac{\boldsymbol{\mathrm{dx}}}{\sqrt{\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{sinx}}\right)\left(\boldsymbol{\mathrm{a}}+\boldsymbol{\mathrm{cosx}}\right)}}\:=? \\ $$$$\left[\boldsymbol{\mathrm{a}}\in\boldsymbol{\mathrm{R}}^{+} ,\boldsymbol{\mathrm{test}}\:\boldsymbol{\mathrm{for}}:\:\:\boldsymbol{\mathrm{a}}=\mathrm{1}\right] \\ $$

Question Number 76955 Answers: 0 Comments: 0

∫ (x^p /(√(1±x^q ))) dx=? [p+q=2n∈N,p<q]

$$\int\:\:\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{p}}} }{\sqrt{\mathrm{1}\pm\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{q}}} }}\:\boldsymbol{\mathrm{dx}}=?\:\:\:\:\left[\boldsymbol{\mathrm{p}}+\boldsymbol{\mathrm{q}}=\mathrm{2}\boldsymbol{\mathrm{n}}\in\boldsymbol{\mathrm{N}},\boldsymbol{\mathrm{p}}<\boldsymbol{\mathrm{q}}\right] \\ $$

Question Number 76954 Answers: 0 Comments: 0

∫ (dx/(√(1+sinx.sin2x))) =?

$$\int\:\:\:\frac{\boldsymbol{\mathrm{dx}}}{\sqrt{\mathrm{1}+\boldsymbol{\mathrm{sinx}}.\boldsymbol{\mathrm{sin}}\mathrm{2}\boldsymbol{\mathrm{x}}}}\:\:=? \\ $$

Question Number 76952 Answers: 1 Comments: 0

show that (√(1+ 2017×2018×2019×2020 )) ∈ N

$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\sqrt{\mathrm{1}+\:\mathrm{2017}×\mathrm{2018}×\mathrm{2019}×\mathrm{2020}\:}\:\in\:\mathbb{N} \\ $$

Question Number 76949 Answers: 0 Comments: 0

Question Number 76948 Answers: 1 Comments: 0

Question Number 76941 Answers: 0 Comments: 4

We usually have to write the date in this form 01/01/2020 to mean the 1^(st) january 2020 What is the first date that is written in this form with eight different figures ? an example : 25/09/1873 “i wish you a sweet and happy new year to all of you”

$$\mathrm{We}\:\mathrm{usually}\:\mathrm{have}\:\mathrm{to}\:\mathrm{write}\:\mathrm{the}\:\mathrm{date}\:\mathrm{in}\:\mathrm{this}\:\mathrm{form}\:\mathrm{01}/\mathrm{01}/\mathrm{2020} \\ $$$$\mathrm{to}\:\mathrm{mean}\:\mathrm{the}\:\mathrm{1}^{\mathrm{st}} \:\mathrm{january}\:\mathrm{2020}\: \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{first}\:\mathrm{date}\:\mathrm{that}\:\mathrm{is}\:\mathrm{written}\:\mathrm{in}\:\mathrm{this}\:\mathrm{form}\:\mathrm{with}\:\mathrm{eight}\:\mathrm{different}\:\mathrm{figures}\:? \\ $$$$\mathrm{an}\:\mathrm{example}\::\:\mathrm{25}/\mathrm{09}/\mathrm{1873}\:\: \\ $$$$``\mathrm{i}\:\mathrm{wish}\:\mathrm{you}\:\mathrm{a}\:\mathrm{sweet}\:\mathrm{and}\:\mathrm{happy}\:\mathrm{new}\:\mathrm{year}\:\mathrm{to}\:\mathrm{all}\:\mathrm{of}\:\mathrm{you}'' \\ $$

Question Number 78027 Answers: 1 Comments: 1