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Question Number 85999 Answers: 0 Comments: 5

∫_0 ^∞ ((sinx^2 )/(1+x^4 ))dx=0.4009 prove that

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx}=\mathrm{0}.\mathrm{4009} \\ $$$${prove}\:{that} \\ $$

Question Number 85985 Answers: 0 Comments: 0

Consider the functionf defined by parf(x) = −x + ((ln x)/x) in the interval : ]0,+∞[. (C_f ) is its representative curve in an orthonormal reference system (O,i^→ ,j^→ ). Calculate lim_(x→0^+ ) f(x), lim_(x→+∞) f(x).

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{function}{f}\:\mathrm{defined}\:\mathrm{by}\:\mathrm{par}{f}\left({x}\right)\:=\:−{x}\:+\:\frac{\mathrm{ln}\:{x}}{{x}}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval} \\ $$$$\left.:\:\right]\mathrm{0},+\infty\left[.\:\:\left({C}_{{f}} \right)\:\mathrm{is}\:\mathrm{its}\:\mathrm{representative}\:\mathrm{curve}\:\mathrm{in}\:\mathrm{an}\:\mathrm{orthonormal}\right. \\ $$$$\mathrm{reference}\:\mathrm{system}\:\left(\mathrm{O},\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right). \\ $$$$\:\mathrm{Calculate}\:\:\underset{{x}\rightarrow\mathrm{0}^{+} \:} {\mathrm{lim}}\:{f}\left({x}\right),\:\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\:{f}\left({x}\right). \\ $$

Question Number 86063 Answers: 1 Comments: 0

Question Number 85982 Answers: 0 Comments: 2

A primitive of the function defned by f(x) = x −1 + (1/(x+1)) is A. F(x) = (x^2 /2) −x + ln(x + 1) B. F(x) = (x^2 /2) + ln(x−1) C. F(x) = (x^2 /2)−x + ln(1−x) D. F(x) = −x + ln(x−1)

$$\mathrm{A}\:\mathrm{primitive}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:\mathrm{defned}\:\mathrm{by}\:\mathrm{f}\left({x}\right)\:=\:{x}\:−\mathrm{1}\:+\:\frac{\mathrm{1}}{{x}+\mathrm{1}}\:\mathrm{is}\: \\ $$$$\mathrm{A}.\:\mathrm{F}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:−{x}\:+\:\mathrm{ln}\left({x}\:+\:\mathrm{1}\right)\:\:\:\:\mathrm{B}.\:\mathrm{F}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}\:+\:\mathrm{ln}\left({x}−\mathrm{1}\right) \\ $$$$\mathrm{C}.\:\mathrm{F}\left({x}\right)\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−{x}\:+\:\mathrm{ln}\left(\mathrm{1}−{x}\right)\:\:\:\:\:\:\:\:\:\mathrm{D}.\:\mathrm{F}\left({x}\right)\:=\:−{x}\:+\:\mathrm{ln}\left({x}−\mathrm{1}\right) \\ $$$$ \\ $$

Question Number 85961 Answers: 0 Comments: 4

Question Number 85952 Answers: 1 Comments: 0

∫ ((√x)/(2+(x)^(1/(3 )) )) dx

$$\int\:\:\frac{\sqrt{\mathrm{x}}}{\mathrm{2}+\sqrt[{\mathrm{3}\:\:}]{\mathrm{x}}}\:\mathrm{dx}\: \\ $$

Question Number 85950 Answers: 1 Comments: 1

Question Number 85935 Answers: 0 Comments: 7

Question Number 85931 Answers: 0 Comments: 0

Question Number 85930 Answers: 0 Comments: 1

Question Number 85915 Answers: 1 Comments: 0

3^(^(∣x∣) log 27) ≥ ((81)/x)

$$\mathrm{3}^{\:^{\mid\mathrm{x}\mid} \mathrm{log}\:\mathrm{27}} \:\geqslant\:\frac{\mathrm{81}}{\mathrm{x}} \\ $$

Question Number 85914 Answers: 1 Comments: 0

Let I_n =∫_(0 ) ^(π/4) (((1−tan A)/(1+tan A)))^n dA what is the Laplace Transform and the Fourier Transform

$${Let}\:{I}_{{n}} =\overset{\pi/\mathrm{4}} {\int}_{\mathrm{0}\:} \left(\frac{\mathrm{1}−\mathrm{tan}\:{A}}{\mathrm{1}+\mathrm{tan}\:{A}}\right)^{{n}} {dA}\:\:{what}\:{is} \\ $$$${the}\:{Laplace}\:{Transform}\:{and}\:{the} \\ $$$${Fourier}\:{Transform} \\ $$

Question Number 85909 Answers: 2 Comments: 2

∫ sin^(−1) ((√(x/(a+x)))) dx , a > 0

$$\int\:\mathrm{sin}^{−\mathrm{1}} \:\left(\sqrt{\frac{\mathrm{x}}{\mathrm{a}+\mathrm{x}}}\right)\:\mathrm{dx}\:,\:\mathrm{a}\:>\:\mathrm{0} \\ $$

Question Number 85902 Answers: 1 Comments: 15

find the coefficients of x^2 and x^3 terms in the expansion of (1+x)(1+2x)^2 (1+3x)^3 ...(1+100x)^(100)

$${find}\:{the}\:{coefficients}\:{of}\:{x}^{\mathrm{2}} \:{and}\:{x}^{\mathrm{3}} \: \\ $$$${terms}\:{in}\:{the}\:{expansion}\:{of} \\ $$$$\left(\mathrm{1}+{x}\right)\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} \left(\mathrm{1}+\mathrm{3}{x}\right)^{\mathrm{3}} ...\left(\mathrm{1}+\mathrm{100}{x}\right)^{\mathrm{100}} \\ $$

Question Number 85896 Answers: 2 Comments: 4

Question Number 85890 Answers: 0 Comments: 0

Is there a sum for ψ((p/q)) or an integral for any fraction p/q

$${Is}\:{there}\:{a}\:{sum}\:{for}\:\psi\left(\frac{{p}}{{q}}\right)\:{or}\:{an}\:{integral} \\ $$$${for}\:{any}\:{fraction}\:{p}/{q} \\ $$

Question Number 85888 Answers: 0 Comments: 5

i−∫_0 ^5 (x+[2x])^([(x/3)]) dx ii−∫_0 ^3 (z−{z})^([z]) dz

$${i}−\int_{\mathrm{0}} ^{\mathrm{5}} \left({x}+\left[\mathrm{2}{x}\right]\right)^{\left[\frac{{x}}{\mathrm{3}}\right]} {dx} \\ $$$$ \\ $$$${ii}−\int_{\mathrm{0}} ^{\mathrm{3}} \left({z}−\left\{{z}\right\}\right)^{\left[{z}\right]} \:{dz} \\ $$

Question Number 85875 Answers: 1 Comments: 0

∫((1/(7[1−(1/7)e^x ]))) dx

$$\int\left(\frac{\mathrm{1}}{\mathrm{7}\left[\mathrm{1}−\frac{\mathrm{1}}{\mathrm{7}}\mathrm{e}^{\mathrm{x}} \right]}\right)\:\mathrm{dx} \\ $$

Question Number 85872 Answers: 1 Comments: 1

∫cos^(2020) x dx = ?

$$\int\mathrm{cos}^{\mathrm{2020}} \mathrm{x}\:\mathrm{dx}\:=\:? \\ $$

Question Number 85871 Answers: 0 Comments: 3

Question Number 85868 Answers: 0 Comments: 6

if f(x)=⌊x^2 ⌋ and A=lim_(x→0) (f(x)−f(−x)) and B=f(x)+f(−x) when x=0 find A and B

$${if}\:{f}\left({x}\right)=\lfloor{x}^{\mathrm{2}} \rfloor\:\: \\ $$$${and}\:{A}=\underset{{x}\rightarrow\mathrm{0}} {{lim}}\left({f}\left({x}\right)−{f}\left(−{x}\right)\right) \\ $$$${and}\:{B}={f}\left({x}\right)+{f}\left(−{x}\right)\:\:{when}\:{x}=\mathrm{0} \\ $$$$ \\ $$$${find}\:{A}\:{and}\:{B} \\ $$

Question Number 85866 Answers: 2 Comments: 1

Question Number 85865 Answers: 1 Comments: 0

prove that curl(r^n c^→ ×r^→ )=(n+2)r^n c^→ −nr^(n−2) (r^→ .c^→ ) . where c is the constant vector.

$$\mathrm{prove}\:\mathrm{that}\: \\ $$$$\mathrm{curl}\left(\mathrm{r}^{\mathrm{n}} \overset{\rightarrow} {\mathrm{c}}×\overset{\rightarrow} {\mathrm{r}}\right)=\left(\mathrm{n}+\mathrm{2}\right)\mathrm{r}^{\mathrm{n}} \overset{\rightarrow} {\mathrm{c}}−\mathrm{nr}^{\mathrm{n}−\mathrm{2}} \left(\overset{\rightarrow} {\mathrm{r}}.\overset{\rightarrow} {\mathrm{c}}\right)\:\:. \\ $$$$\mathrm{where}\:\mathrm{c}\:\mathrm{is}\:\mathrm{the}\:\mathrm{constant}\:\mathrm{vector}. \\ $$

Question Number 85864 Answers: 1 Comments: 0

simplify the expression (√(6+2(√(8(√3)−10)))) − (√(7−(√3))) in the form (√((√a)+b)) ?

$${simplify}\:{the}\:{expression} \\ $$$$\sqrt{\mathrm{6}+\mathrm{2}\sqrt{\mathrm{8}\sqrt{\mathrm{3}}−\mathrm{10}}}\:−\:\sqrt{\mathrm{7}−\sqrt{\mathrm{3}}}\:\:{in} \\ $$$${the}\:{form}\:\sqrt{\sqrt{{a}}+{b}}\:? \\ $$

Question Number 85859 Answers: 2 Comments: 0

Is the Var(aX+b) = a^2 Var(X) + b?

$$\:\mathrm{Is}\:\mathrm{the}\:\mathrm{Var}\left(\mathrm{aX}+\mathrm{b}\right)\:=\:\mathrm{a}^{\mathrm{2}} \:\mathrm{Var}\left(\mathrm{X}\right)\:+\:\mathrm{b}? \\ $$

Question Number 85858 Answers: 0 Comments: 0

Is a matrix A^T A always positive definite?

$$\mathrm{Is}\:\mathrm{a}\:\mathrm{matrix} \\ $$$$\mathrm{A}^{\mathrm{T}} \mathrm{A}\:\mathrm{always}\:\mathrm{positive}\:\mathrm{definite}? \\ $$

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