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AllQuestion and Answers: Page 125

Question Number 210996 Answers: 2 Comments: 0

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Question Number 210989 Answers: 1 Comments: 0

prove tan(72^° )=tan(66^° )+tan(36^° )+tan(6^° )

$${prove}\:{tan}\left(\mathrm{72}^{°} \right)={tan}\left(\mathrm{66}^{°} \right)+{tan}\left(\mathrm{36}^{°} \right)+{tan}\left(\mathrm{6}^{°} \right)\:\: \\ $$

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(√(25))

$$\sqrt{\mathrm{25}} \\ $$

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Q.210956 im read leithold book again , in this book : 1}define : ln(x)=∫_1 ^( x) dx/x x>0 2}define : ln(e)=1=∫_1 ^( e) dx/x 3}define : exp(x)=y ⇔ ln(y)=x ((d(ln(u)))/du)=(1/u) ⇒ ((d(ln(u)))/dx)=((du/dx)/u) ⇒ u=x^r ⇒ ((d(ln(x^r )))/dx)=((rx^(r−1) )/x^r )=r×(1/x)=r×((d(ln(x)))/dx) ⇒ ln(x^r )=rln(x)+K ⇒ x=1 ⇒ K=0 ⇒ ln(x^r )=r×ln(x) ∀x>0 , ∀r get x=e ⇒ ln(e^r )=r×ln(e)=r ⇒ exp(r)=e^r ⇒ exp(x)=e^x =y ∀x log_e (e^x )=log_e (y)=x and define: ln(y)=x ⇒⇒⇒ln(y)=log_e (y)=x

$${Q}.\mathrm{210956} \\ $$$${im}\:{read}\:{leithold}\:{book}\:{again}\:,\:{in}\:{this}\:{book}\:: \\ $$$$\left.\mathrm{1}\right\}{define}\::\:{ln}\left({x}\right)=\int_{\mathrm{1}} ^{\:{x}} {dx}/{x}\:\:\:\:\:\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right\}{define}\::\:{ln}\left({e}\right)=\mathrm{1}=\int_{\mathrm{1}} ^{\:{e}} {dx}/{x} \\ $$$$\left.\mathrm{3}\right\}{define}\::\:{exp}\left({x}\right)={y}\:\Leftrightarrow\:{ln}\left({y}\right)={x} \\ $$$$\frac{{d}\left({ln}\left({u}\right)\right)}{{du}}=\frac{\mathrm{1}}{{u}}\:\Rightarrow\:\frac{{d}\left({ln}\left({u}\right)\right)}{{dx}}=\frac{{du}/{dx}}{{u}}\:\Rightarrow \\ $$$${u}={x}^{{r}} \:\Rightarrow\:\frac{{d}\left({ln}\left({x}^{{r}} \right)\right)}{{dx}}=\frac{{rx}^{{r}−\mathrm{1}} }{{x}^{{r}} }={r}×\frac{\mathrm{1}}{{x}}={r}×\frac{{d}\left({ln}\left({x}\right)\right)}{{dx}} \\ $$$$\Rightarrow\:{ln}\left({x}^{{r}} \right)={rln}\left({x}\right)+{K}\:\Rightarrow\:{x}=\mathrm{1}\:\Rightarrow\:{K}=\mathrm{0} \\ $$$$\Rightarrow\:{ln}\left({x}^{{r}} \right)={r}×{ln}\left({x}\right)\:\:\:\:\:\:\forall{x}>\mathrm{0}\:,\:\forall{r} \\ $$$${get}\:\:{x}={e}\:\:\Rightarrow\:{ln}\left({e}^{{r}} \right)={r}×{ln}\left({e}\right)={r}\:\Rightarrow\:{exp}\left({r}\right)={e}^{{r}} \\ $$$$\Rightarrow\:{exp}\left({x}\right)={e}^{{x}} ={y}\:\:\:\:\:\forall{x} \\ $$$${log}_{{e}} \left({e}^{{x}} \right)={log}_{{e}} \left({y}\right)={x}\:\:\:{and}\:\:\:{define}:\:{ln}\left({y}\right)={x} \\ $$$$\Rightarrow\Rightarrow\Rightarrow{ln}\left({y}\right)={log}_{{e}} \left({y}\right)={x} \\ $$

Question Number 210972 Answers: 2 Comments: 0

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if a_n = n^4 ∫_n ^(n+1) ((x dx)/(1+x^5 )) then (1) Σa_n is convergent or divergent?? (2) lim_(n→∞) a_(n ) = ??

$$\:\:\:\:\mathrm{if}\:\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{n}^{\mathrm{4}} \int_{\mathrm{n}} ^{\mathrm{n}+\mathrm{1}} \:\frac{\mathrm{x}\:\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{5}} }\:\:\mathrm{then} \\ $$$$\:\:\:\:\left(\mathrm{1}\right)\:\Sigma\mathrm{a}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{convergent}\:\mathrm{or}\:\mathrm{divergent}?? \\ $$$$\:\:\:\:\left(\mathrm{2}\right)\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\mathrm{n}\:} \:=\:?? \\ $$

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Question Number 210956 Answers: 1 Comments: 1

we define : ln(x)=∫_1 ^( x) (dx/x) how prove : ln(x)=log_e x ?

$${we}\:{define}\::\:{ln}\left({x}\right)=\int_{\mathrm{1}} ^{\:{x}} \frac{{dx}}{{x}} \\ $$$${how}\:{prove}\::\:{ln}\left({x}\right)={log}_{{e}} {x}\:\:\:? \\ $$

Question Number 210948 Answers: 2 Comments: 1

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∫_0 ^1 ((log(x)tanh^(−1) (x)log(x^2 +1))/x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{log}\left({x}\right){tanh}^{−\mathrm{1}} \left({x}\right){log}\left({x}^{\mathrm{2}} +\mathrm{1}\right)}{{x}}{dx} \\ $$

Question Number 210935 Answers: 2 Comments: 0

at what times, if exist, are the angles betwen the hour hand, the minute hand and the second hand of a clock exactly 120°? assume that the hands of the clock move uniformly.

$${at}\:{what}\:{times},\:{if}\:{exist},\:{are}\:{the}\: \\ $$$${angles}\:{betwen}\:{the}\:{hour}\:{hand},\:{the} \\ $$$${minute}\:{hand}\:{and}\:{the}\:{second}\:{hand} \\ $$$${of}\:{a}\:{clock}\:{exactly}\:\mathrm{120}°? \\ $$$${assume}\:{that}\:{the}\:{hands}\:{of}\:{the}\:{clock} \\ $$$${move}\:{uniformly}. \\ $$

Question Number 210934 Answers: 2 Comments: 0

If ((x^2 + ax − 18)/(x^2 + 7x + 2b)) = ((x − c)/(x + 5)) Find a + b + c = ?

$$\mathrm{If}\:\:\:\frac{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{ax}\:−\:\mathrm{18}}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{7x}\:+\:\mathrm{2b}}\:\:=\:\:\frac{\mathrm{x}\:−\:\mathrm{c}}{\mathrm{x}\:+\:\mathrm{5}} \\ $$$$\mathrm{Find}\:\:\:\boldsymbol{\mathrm{a}}\:+\:\boldsymbol{\mathrm{b}}\:+\:\boldsymbol{\mathrm{c}}\:=\:? \\ $$

Question Number 210933 Answers: 1 Comments: 0

I= ∫_0 ^( (π/2) ) (( sin( 25x ))/(sinx)) dx=?

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}\:} \frac{\:{sin}\left(\:\mathrm{25}{x}\:\right)}{{sinx}}\:{dx}=? \\ $$$$ \\ $$$$ \\ $$

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