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Question Number 214560 Answers: 1 Comments: 0

∫_1 ^( x) ((ln x)/( (√(1−(ln x)^2 ))))dx

$$\int_{\mathrm{1}} ^{\:\:{x}} \frac{\mathrm{ln}\:{x}}{\:\sqrt{\mathrm{1}−\left(\mathrm{ln}\:{x}\right)^{\mathrm{2}} }}{dx} \\ $$

Question Number 214514 Answers: 0 Comments: 7

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Question Number 214511 Answers: 0 Comments: 0

Let′s R(z) define as R(z)=((π ∫_0 ^( z) f^2 (t)dt)/(2π ∫_0 ^( z) f(t)(√(1+(f^((1)) (t))^2 ))dt)) and both integral ∫_0 ^( ∞) f^( 2) (t)dt , ∫_0 ^( ∞) f(t)(√(1+(f^((1)) (t))^2 ))dt =∞ lim_(z→∞) ((π ∫_0 ^( z) f^( 2) (t)dt)/(2π ∫_0 ^( z) f(t)(√(1+(f^((1)) (t))^2 ))dt)) =lim_(z→∞) ((π f(z))/(2π (√(1+(f^((1)) (z))^2 )))) ..??

$$\mathrm{Let}'\mathrm{s}\:{R}\left({z}\right)\:\mathrm{define}\:\mathrm{as}\: \\ $$$${R}\left({z}\right)=\frac{\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}^{\mathrm{2}} \left({t}\right)\mathrm{d}{t}}{\mathrm{2}\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}} \\ $$$$\mathrm{and}\:\mathrm{both}\:\mathrm{integral} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:{f}^{\:\mathrm{2}} \left({t}\right)\mathrm{d}{t}\:,\:\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}\:=\infty \\ $$$$\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi\:\int_{\mathrm{0}} ^{\:{z}} {f}^{\:\mathrm{2}} \left({t}\right)\mathrm{d}{t}}{\mathrm{2}\pi\:\int_{\mathrm{0}} ^{\:{z}} \:{f}\left({t}\right)\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({t}\right)\right)^{\mathrm{2}} }\mathrm{d}{t}} \\ $$$$=\underset{{z}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi\:{f}\left({z}\right)}{\mathrm{2}\pi\:\sqrt{\mathrm{1}+\left({f}^{\left(\mathrm{1}\right)} \left({z}\right)\right)^{\mathrm{2}} }}\:..?? \\ $$

Question Number 214509 Answers: 1 Comments: 4

Find the volume of the solid of revolution generated by rotating the area bounded by y=x(2−x) and y=x about the y−axis.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{volume}\:\mathrm{of}\:\mathrm{the}\:\mathrm{solid}\:\mathrm{of}\:\mathrm{revolution} \\ $$$$\mathrm{generated}\:\mathrm{by}\:\mathrm{rotating}\:\mathrm{the}\:\mathrm{area}\:\mathrm{bounded} \\ $$$$\mathrm{by}\:{y}={x}\left(\mathrm{2}−{x}\right)\:\mathrm{and}\:{y}={x}\:\mathrm{about}\:\mathrm{the}\:\mathrm{y}−\mathrm{axis}. \\ $$

Question Number 214499 Answers: 3 Comments: 1

{ ((x^2 + (x + 3y) = 11)),((y^2 + (y + 3x) = 29)) :} ⇒ x + y = ?

$$\begin{cases}{\mathrm{x}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{x}\:\:+\:\:\mathrm{3y}\right)\:\:=\:\:\mathrm{11}}\\{\mathrm{y}^{\mathrm{2}} \:\:+\:\:\left(\mathrm{y}\:\:+\:\:\mathrm{3x}\right)\:\:=\:\:\mathrm{29}}\end{cases}\:\:\:\:\:\Rightarrow\:\:\:\:\mathrm{x}\:+\:\mathrm{y}\:=\:? \\ $$

Question Number 214498 Answers: 1 Comments: 0

Question Number 214495 Answers: 0 Comments: 1

Question Number 214485 Answers: 1 Comments: 0

lim_(n→∞) ((1/(2∙4))+(1/(5∙7))+...+(1/((3n−1)(3n+1))))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{2}\centerdot\mathrm{4}}+\frac{\mathrm{1}}{\mathrm{5}\centerdot\mathrm{7}}+...+\frac{\mathrm{1}}{\left(\mathrm{3}{n}−\mathrm{1}\right)\left(\mathrm{3}{n}+\mathrm{1}\right)}\right) \\ $$

Question Number 214569 Answers: 2 Comments: 0

Question Number 214568 Answers: 1 Comments: 0

∫((b+ax)/(1+sin x)) dx

$$\int\frac{{b}+{ax}}{\mathrm{1}+\mathrm{sin}\:{x}}\:{dx} \\ $$

Question Number 214566 Answers: 2 Comments: 0

somebody has posted following question and then deleted it again. { ((u_(n+1) =((4u_n −9)/(u_n −2)))),((u_0 =5)) :} find u_n =? (or something like this)

$${somebody}\:{has}\:{posted}\:{following} \\ $$$${question}\:{and}\:{then}\:{deleted}\:{it}\:{again}. \\ $$$$\begin{cases}{{u}_{{n}+\mathrm{1}} =\frac{\mathrm{4}{u}_{{n}} −\mathrm{9}}{{u}_{{n}} −\mathrm{2}}}\\{{u}_{\mathrm{0}} =\mathrm{5}}\end{cases} \\ $$$${find}\:{u}_{{n}} =?\:\left({or}\:{something}\:{like}\:{this}\right) \\ $$

Question Number 214483 Answers: 1 Comments: 0