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Question Number 206808 Answers: 2 Comments: 0

lim_(x→0) ((10^x −1)/x^(10) )

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{10}^{\mathrm{x}} −\mathrm{1}}{\mathrm{x}^{\mathrm{10}} } \\ $$

Question Number 206806 Answers: 0 Comments: 0

Given is a square with side length 15. We need to find exactly 17 smaller squares to fill the big one. How many solutions are possible? (Note: it′s not enough to find squares with the sum of their areas being 225, they must fit into the 15×15 square. Example with 3 squares: 2×2+5×5+14×14=225 but you cannot fit these in a 15×15 square)

$$\mathrm{Given}\:\mathrm{is}\:\mathrm{a}\:\mathrm{square}\:\mathrm{with}\:\mathrm{side}\:\mathrm{length}\:\mathrm{15}. \\ $$$$\mathrm{We}\:\mathrm{need}\:\mathrm{to}\:\mathrm{find}\:\mathrm{exactly}\:\mathrm{17}\:\mathrm{smaller}\:\mathrm{squares} \\ $$$$\mathrm{to}\:\mathrm{fill}\:\mathrm{the}\:\mathrm{big}\:\mathrm{one}.\:\mathrm{How}\:\mathrm{many}\:\mathrm{solutions}\:\mathrm{are} \\ $$$$\mathrm{possible}? \\ $$$$\left(\mathrm{Note}:\:\mathrm{it}'\mathrm{s}\:\mathrm{not}\:\mathrm{enough}\:\mathrm{to}\:\mathrm{find}\:\mathrm{squares}\:\mathrm{with}\right. \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{their}\:\mathrm{areas}\:\mathrm{being}\:\mathrm{225},\:\mathrm{they}\:\mathrm{must} \\ $$$$\mathrm{fit}\:\mathrm{into}\:\mathrm{the}\:\mathrm{15}×\mathrm{15}\:\mathrm{square}.\:\mathrm{Example}\:\mathrm{with} \\ $$$$\mathrm{3}\:\mathrm{squares}:\:\mathrm{2}×\mathrm{2}+\mathrm{5}×\mathrm{5}+\mathrm{14}×\mathrm{14}=\mathrm{225}\:\mathrm{but} \\ $$$$\left.\mathrm{you}\:\mathrm{cannot}\:\mathrm{fit}\:\mathrm{these}\:\mathrm{in}\:\mathrm{a}\:\mathrm{15}×\mathrm{15}\:\mathrm{square}\right) \\ $$

Question Number 206805 Answers: 2 Comments: 0

Question Number 206804 Answers: 2 Comments: 0

Question Number 206795 Answers: 0 Comments: 3

Question Number 206794 Answers: 1 Comments: 0

help me... ∫_0 ^∞ ((sin(t)ln(t))/t)e^(−t) dt

$${help}\:{me}... \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{sin}\left({t}\right)\mathrm{ln}\left({t}\right)}{{t}}{e}^{−{t}} \:{dt} \\ $$

Question Number 206787 Answers: 0 Comments: 1

Question Number 206789 Answers: 1 Comments: 0

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Question Number 206773 Answers: 0 Comments: 3

∫_0 ^∞ (e^(−x^2 ) /((x^2 +(1/2))^2 ))dx= I^2 =∫∫_( D) (e^(−x^2 −y^2 ) /((x^2 +(1/2))^2 (y^2 +(1/2))^2 ))dA x=rcos(θ) y=rsin(θ) J=∣((∂(x,y))/(∂(r,θ)))∣drdθ=rdrdθ ∫∫_( D) ((re^(−r^2 ) )/((r^2 cos^2 (θ)+(1/2))^2 (r^2 sin^2 (θ)+(1/2))^2 ))drdθ

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{e}^{−{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{dx}= \\ $$$${I}^{\mathrm{2}} =\int\int_{\:\boldsymbol{\mathcal{D}}} \:\frac{{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \left({y}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{dA} \\ $$$${x}={r}\mathrm{cos}\left(\theta\right)\:\:{y}={r}\mathrm{sin}\left(\theta\right) \\ $$$${J}=\mid\frac{\partial\left({x},{y}\right)}{\partial\left({r},\theta\right)}\mid{drd}\theta={rdrd}\theta \\ $$$$\int\int_{\:\boldsymbol{\mathcal{D}}} \:\frac{{re}^{−{r}^{\mathrm{2}} } }{\left({r}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \left(\theta\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \left({r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)+\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} }{drd}\theta \\ $$

Question Number 206764 Answers: 2 Comments: 0