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Question Number 221957    Answers: 2   Comments: 0

∫sin^(−1) (cos x)dx

$$\int\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{cos}\:{x}\right){dx} \\ $$

Question Number 221955    Answers: 0   Comments: 0

∫_0 ^∞ tan^(−1) (((ln(sin (x))/x)) dx

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{ln}\left(\mathrm{sin}\:\left({x}\right)\right.}{{x}}\right)\:\mathrm{d}{x} \\ $$$$ \\ $$

Question Number 221944    Answers: 1   Comments: 1

Question Number 221943    Answers: 1   Comments: 0

Question Number 221939    Answers: 2   Comments: 1

sin^2 1° + sin^2 3° + sin^2 5° + ... + sin^2 269° = ? Help me, please

$$ \\ $$$$\:\:\:{sin}^{\mathrm{2}} \mathrm{1}°\:+\:{sin}^{\mathrm{2}} \mathrm{3}°\:+\:{sin}^{\mathrm{2}} \mathrm{5}°\:+\:...\:+\:{sin}^{\mathrm{2}} \mathrm{269}°\:=\:? \\ $$$$\:\:\:\mathcal{H}{elp}\:{me},\:\:{please} \\ $$$$ \\ $$

Question Number 221919    Answers: 0   Comments: 6

Acable can stand a maximum weight of 25kg If the length of wire is halved what is the maximum weight thsn can be hung from it??

$${Acable}\:{can}\:{stand}\:{a}\:{maximum}\:{weight}\:{of}\:\mathrm{25kg}\: \\ $$$${If}\:{the}\:{length}\:{of}\:{wire}\:{is}\:{halved}\:{what}\:{is}\:{the} \\ $$$${maximum}\:{weight}\:{thsn}\:{can}\:{be}\:{hung}\:{from}\:{it}?? \\ $$

Question Number 221913    Answers: 1   Comments: 0

A man standing 10m away from a plane mirror moves towards the mirror with a speed of 2m/s. Calculate (a) The speed with which his image in the mirrow approaches him (b) How far is the man from his image after 3 seconds.

A man standing 10m away from a plane mirror moves towards the mirror with a speed of 2m/s. Calculate (a) The speed with which his image in the mirrow approaches him (b) How far is the man from his image after 3 seconds.

Question Number 221907    Answers: 2   Comments: 0

Question Number 221902    Answers: 2   Comments: 0

Question Number 221899    Answers: 0   Comments: 0

let a∈[0,1] find all continuous function f:[0,1]→[0,∞) such that ∫_0 ^1 f(x)dx = 1 , ∫_0 ^1 xf(x)dx = a and ∫_0 ^1 x^2 f(x)dx = a^2 how many such function are there ?

$$\:\:\:{let}\:{a}\in\left[\mathrm{0},\mathrm{1}\right]\:{find}\:{all}\:{continuous}\:{function} \\ $$$$\:\:\:\:\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\left[\mathrm{0},\infty\right)\:{such}\:{that}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx}\:=\:\mathrm{1}\:\:, \\ $$$$\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} {xf}\left({x}\right){dx}\:=\:{a}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} {f}\left({x}\right){dx}\:=\:{a}^{\mathrm{2}} \\ $$$$\:\:\:\:{how}\:{many}\:{such}\:{function}\:{are}\:{there}\:? \\ $$$$ \\ $$

Question Number 221896    Answers: 1   Comments: 0

If log _(10) 7=a ,then log _(10) ((1/(70)))=?

$${If}\:\mathrm{log}\underset{\mathrm{10}} {\:}\mathrm{7}={a}\:,{then}\:\mathrm{log}\underset{\mathrm{10}} {\:}\left(\frac{\mathrm{1}}{\mathrm{70}}\right)=? \\ $$

Question Number 221882    Answers: 1   Comments: 0

Prove:∫_0 ^(π/2) sin x K^2 sin x dx=(π^4 /(16)) _7 F_6 ((1/( 2)),(1/2),(1/2),(1/2),(1/2),(1/2),(5/4);1,1,1,1,1,(1/4);1)

$$\mathrm{Prove}:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{sin}\:{x}\:{K}^{\mathrm{2}} \:\mathrm{sin}\:{x}\:{dx}=\frac{\pi^{\mathrm{4}} }{\mathrm{16}}\:_{\mathrm{7}} {F}_{\mathrm{6}} \left(\frac{\mathrm{1}}{\:\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{1}}{\mathrm{2}},\frac{\mathrm{5}}{\mathrm{4}};\mathrm{1},\mathrm{1},\mathrm{1},\mathrm{1},\mathrm{1},\frac{\mathrm{1}}{\mathrm{4}};\mathrm{1}\right) \\ $$

Question Number 221870    Answers: 2   Comments: 0

Question Number 221869    Answers: 1   Comments: 0

if a^(3−x) .b^(5x) =a^(5+x) .b^(3x) then show that xlog ((b/a))=log a

$${if}\:{a}^{\mathrm{3}−{x}} .{b}^{\mathrm{5}{x}} ={a}^{\mathrm{5}+{x}} .{b}^{\mathrm{3}{x}} \:{then}\:{show}\:{that} \\ $$$${x}\mathrm{log}\:\left(\frac{{b}}{{a}}\right)=\mathrm{log}\:{a} \\ $$

Question Number 221863    Answers: 1   Comments: 1

If a and b are whole numbers such a^b =121 then find the value of (a−1)^(b+1)

$${If}\:{a}\:{and}\:{b}\:{are}\:{whole}\:{numbers}\:{such}\:{a}^{{b}} =\mathrm{121} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\left({a}−\mathrm{1}\right)^{{b}+\mathrm{1}} \\ $$

Question Number 221856    Answers: 0   Comments: 3

Question Number 221853    Answers: 2   Comments: 3

find x where log _8 x−log _4 x−log _2 x=11

$${find}\:{x}\:{where} \\ $$$$\mathrm{log}\underset{\mathrm{8}} {\:}{x}−\mathrm{log}\underset{\mathrm{4}} {\:}{x}−\mathrm{log}\underset{\mathrm{2}} {\:}{x}=\mathrm{11} \\ $$

Question Number 221848    Answers: 0   Comments: 2

Question Number 221843    Answers: 0   Comments: 2

∫_0 ^( (π/2)) x^2 csc^2 (x)dx ∫ −((4z^2 )/((e^(iz) −e^(−iz) )^2 )) dz=∫ ((z^2 e^(2iz) )/((e^(2iz) −1)^2 )) dz u=^(Substitute) e^(2iz) −1 →du=2ie^(2iz) dz z^2 =−((ln^2 (u+1))/4) −(1/8)i ∫ ((ln^2 (u+1))/u^2 ) du by parts ∫ f(u)g′(u)du=f(u)g(u)−∫ f′(u)g(u)du f(u)=ln^2 (u+1) g′(u)=(1/u) f′(u)=((2ln(u+1))/(u+1)) g(u)=−(1/u) −((ln^2 (u+1))/u)−∫ −((2ln(u+1))/(u(u+1))) du −2∫ ((ln(u+1))/(u(u+1))) du...? ∫ ( ((ln(u+1))/u)−((ln(u+1))/(u+1)))du ∫ ((ln(u+1))/u) du=−∫ −((ln(1−v))/v) dv ∴−Li_2 (−u) ∫ ((ln(u+1))/(u+1)) du=∫ v dv (∵ ln(u+1)=v) (1/2)(ln(u+1))^2 ∫ ((ln(u+1))/u) du−∫ ((ln(u+1))/(u+1)) du= −((ln^2 (u+1))/2)−Li_2 (−u) −2∫ ((ln(u+1))/(u(u+1))) du=ln^2 (u+1)+2Li_2 (−u) −((ln^2 (u+1))/u)−∫ −((2ln(u+1))/(u+1)) du= −((ln^2 (u+1))/u)−ln^2 (u+1)−2Li_2 (−u) plug to solve integrals (1/8)i ∫ ((ln^2 (u+1))/u^2 ) du −((iln^2 (u+1))/(8u))−((iln^2 (u+1))/8)−((iLi_2 (−u))/4) u=e^(2ix) −1 ∴ iLi_2 (1−e^(2ix) )−((2ix^2 )/(e^(2ix) −1))−2ix^2 +Const ∫_0 ^( (π/2)) - =[iLi_2 (1−e^(2ix) )−((2ix^2 )/(e^(2ix) −1))−2ix^2 ]_(x=0) ^(x=(π/2))

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\:{x}^{\mathrm{2}} \mathrm{csc}^{\mathrm{2}} \left({x}\right)\mathrm{d}{x} \\ $$$$\int\:\:−\frac{\mathrm{4}{z}^{\mathrm{2}} }{\left({e}^{\boldsymbol{{i}}{z}} −{e}^{−\boldsymbol{{i}}{z}} \right)^{\mathrm{2}} }\:\mathrm{d}{z}=\int\:\:\frac{{z}^{\mathrm{2}} {e}^{\mathrm{2}\boldsymbol{{i}}{z}} }{\left({e}^{\mathrm{2}\boldsymbol{{i}}{z}} −\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{d}{z} \\ $$$${u}\overset{\mathrm{Substitute}} {=}{e}^{\mathrm{2}\boldsymbol{{i}}{z}} −\mathrm{1}\:\:\rightarrow\mathrm{d}{u}=\mathrm{2}\boldsymbol{{i}}{e}^{\mathrm{2}\boldsymbol{{i}}{z}} \:\mathrm{d}{z} \\ $$$${z}^{\mathrm{2}} =−\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{\mathrm{4}}\: \\ $$$$−\frac{\mathrm{1}}{\mathrm{8}}\boldsymbol{{i}}\:\int\:\:\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{{u}^{\mathrm{2}} }\:\mathrm{d}{u} \\ $$$$\mathrm{by}\:\mathrm{parts}\:\int\:{f}\left({u}\right)\mathrm{g}'\left({u}\right)\mathrm{d}{u}={f}\left({u}\right)\mathrm{g}\left({u}\right)−\int\:{f}'\left({u}\right)\mathrm{g}\left({u}\right)\mathrm{d}{u} \\ $$$${f}\left({u}\right)=\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)\:\mathrm{g}'\left({u}\right)=\frac{\mathrm{1}}{{u}} \\ $$$${f}'\left({u}\right)=\frac{\mathrm{2ln}\left({u}+\mathrm{1}\right)}{{u}+\mathrm{1}}\:\mathrm{g}\left({u}\right)=−\frac{\mathrm{1}}{{u}} \\ $$$$−\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{{u}}−\int\:−\frac{\mathrm{2ln}\left({u}+\mathrm{1}\right)}{{u}\left({u}+\mathrm{1}\right)}\:\mathrm{d}{u} \\ $$$$−\mathrm{2}\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}\left({u}+\mathrm{1}\right)}\:\mathrm{d}{u}...? \\ $$$$\int\:\left(\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}}−\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}+\mathrm{1}}\right)\mathrm{d}{u} \\ $$$$\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}}\:\mathrm{d}{u}=−\int\:−\frac{\mathrm{ln}\left(\mathrm{1}−{v}\right)}{{v}}\:\mathrm{d}{v} \\ $$$$\therefore−\mathrm{Li}_{\mathrm{2}} \left(−{u}\right) \\ $$$$\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}+\mathrm{1}}\:\mathrm{d}{u}=\int\:{v}\:\mathrm{d}{v}\:\left(\because\:\mathrm{ln}\left({u}+\mathrm{1}\right)={v}\right) \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{ln}\left({u}+\mathrm{1}\right)\right)^{\mathrm{2}} \\ $$$$\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}}\:\mathrm{d}{u}−\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}+\mathrm{1}}\:\mathrm{d}{u}= \\ $$$$−\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{\mathrm{2}}−\mathrm{Li}_{\mathrm{2}} \left(−{u}\right) \\ $$$$−\mathrm{2}\int\:\:\frac{\mathrm{ln}\left({u}+\mathrm{1}\right)}{{u}\left({u}+\mathrm{1}\right)}\:\mathrm{d}{u}=\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)+\mathrm{2Li}_{\mathrm{2}} \left(−{u}\right) \\ $$$$−\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{{u}}−\int\:\:−\frac{\mathrm{2ln}\left({u}+\mathrm{1}\right)}{{u}+\mathrm{1}}\:\mathrm{d}{u}= \\ $$$$−\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{{u}}−\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)−\mathrm{2Li}_{\mathrm{2}} \left(−{u}\right) \\ $$$$\mathrm{plug}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{integrals} \\ $$$$\frac{\mathrm{1}}{\mathrm{8}}\boldsymbol{{i}}\:\int\:\:\frac{\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{{u}^{\mathrm{2}} }\:\mathrm{d}{u} \\ $$$$−\frac{\boldsymbol{{i}}\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{\mathrm{8}{u}}−\frac{\boldsymbol{{i}}\mathrm{ln}^{\mathrm{2}} \left({u}+\mathrm{1}\right)}{\mathrm{8}}−\frac{\boldsymbol{{i}}\mathrm{Li}_{\mathrm{2}} \left(−{u}\right)}{\mathrm{4}} \\ $$$${u}={e}^{\mathrm{2}\boldsymbol{{i}}{x}} −\mathrm{1} \\ $$$$\therefore\:\boldsymbol{{i}}\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}−{e}^{\mathrm{2}\boldsymbol{{i}}{x}} \right)−\frac{\mathrm{2}\boldsymbol{{i}}{x}^{\mathrm{2}} }{{e}^{\mathrm{2}\boldsymbol{{i}}{x}} −\mathrm{1}}−\mathrm{2}\boldsymbol{{i}}{x}^{\mathrm{2}} +\mathrm{Const} \\ $$$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:-\:=\left[\boldsymbol{{i}}\mathrm{Li}_{\mathrm{2}} \left(\mathrm{1}−{e}^{\mathrm{2}\boldsymbol{{i}}{x}} \right)−\frac{\mathrm{2}\boldsymbol{{i}}{x}^{\mathrm{2}} }{{e}^{\mathrm{2}\boldsymbol{{i}}{x}} −\mathrm{1}}−\mathrm{2}\boldsymbol{{i}}{x}^{\mathrm{2}} \right]_{{x}=\mathrm{0}} ^{{x}=\frac{\pi}{\mathrm{2}}} \\ $$

Question Number 221838    Answers: 2   Comments: 0

∫_0 ^∞ (((arctan x)/x))^2 dx=?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\left(\frac{\mathrm{arctan}\:{x}}{{x}}\right)^{\mathrm{2}} {dx}=? \\ $$

Question Number 221832    Answers: 0   Comments: 0

f′(x)=lim_(h→0) ((f(x+h)−f(x))/h) lim_(h→0) ((c−c)/h)=lim_(h→0) 0=0 lim_(h→0) (((x+h)^2 −x^2 )/h)=lim_(h→0) ((2xh+h^2 )/h)=lim_(h→0) (2x+h)=2x lim_(h→0) (((x+h)^n −x^n )/h)=lim_(h→0) ((Σ_(k=0) ^n ((n),(k) )x^(n−k) h^k −x^n )/h)=lim_(h→0) ((nx^(n−1) h+ ((n),(2) )x^(n−2) h^2 +…)/h)=lim_(h→0) (nx^(n−1) + ((n),(2) )x^(n−2) h+…) lim_(h→0) ((e^(x+h) −e^x )/h)=e^x lim_(h→0) (e^(h−1) /h)=e^x ∙1 lim_(h→0) ((e^h −1)/h)=lim_(k→0) (k/(ln(1+k)))=lim_(k→0) (1/((ln(1+k))/k))=lim_(k→0) (1/(ln((1+k)^(1/k) ))) =(1/(ln e))=(1/1)=1 lim_(h→0) ((sin(x+h)−sin x)/h)=lim_(h→0) ((sin x cos h+cos x sin h−sin x)/h)=lim_(h→0) (sin x((cos h−1)/h)+cos x((sin h)/h)) lim_(h→0) ((sin h)/h)=1 lim_(h→0) ((cos h−1)/h)=lim_(h→0) ((−2 sin^2 (h/2))/h)=lim_(h→0) −((sin^2 (h/2))/((h/2)^2 ))∙(((h/2)^2 )/h)=lim_(h→0) −(((sin(h/2))/(h/2)))^2 ∙(h/4)=−1∙0=0 lim_(h→0) (sin x∙0+cos x∙1)=cos x ∫_a ^b f(x)dx=lim_(n→∞) Σ_(i=1) ^n f(a+i((b−a)/n))((b−a)/n) ∫_a ^b x^m dx=lim_(n→∞) Σ_(k=1) ^n (k(b/n))^m (b/n)=b^(m+1) lim_(n→∞) (1/n)Σ_(k=1) ^n ((k/n))^m lim_(n→∞) Σ_(k=1) ^n ((k/n))^m (b/n)=lim_(n→∞) (((n^(m+1) /(m+1))+O(n^m ))/n^(m+1) )=lim_(n→∞) ((1/(m+1))+O((1/n)))=(1/(m+1)) ∫_a ^b x^m dx=b^(m+1) ∙(1/(m+1))=(b^(m+1) /(m+1)) ∫x^m dx=(x^(m+1) /(m+1))+C G(x)=∫_0 ^x f(t)dt G′(x)=lim_(h→0) ((G(x+h)−G(x))/h)=lim_(h→0) ((∫_x ^(x+h) f(t)dt)/h)=lim_(h→0) ((f(c_n )h)/h)=lim_(h→0) f(c_n )=f(x) ,c_h ∈[x,x+h] F(b)−F(a)=[∫_a ^b f(t)dt+C]−[∫_a ^a f(t)dt+C]=∫_a ^b f(t)dt−0=∫_a ^b f(t)dt

$${f}'\left({x}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({x}+{h}\right)−{f}\left({x}\right)}{{h}} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{c}−{c}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}0}=\mathrm{0} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{\mathrm{2}} −{x}^{\mathrm{2}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{2}{xh}+{h}^{\mathrm{2}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{2}{x}+{h}\right)=\mathrm{2}{x} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left({x}+{h}\right)^{{n}} −{x}^{{n}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}{x}^{{n}−{k}} {h}^{{k}} −{x}^{{n}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{nx}^{{n}−\mathrm{1}} {h}+\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}{x}^{{n}−\mathrm{2}} {h}^{\mathrm{2}} +\ldots}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left({nx}^{{n}−\mathrm{1}} +\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}{x}^{{n}−\mathrm{2}} {h}+\ldots\right) \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{{x}+{h}} −{e}^{{x}} }{{h}}={e}^{{x}} \underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{{h}−\mathrm{1}} }{{h}}={e}^{{x}} \centerdot\mathrm{1} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{{h}} −\mathrm{1}}{{h}}=\underset{{k}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{k}}{\mathrm{ln}\left(\mathrm{1}+{k}\right)}=\underset{{k}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\frac{\mathrm{ln}\left(\mathrm{1}+{k}\right)}{{k}}}=\underset{{k}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{ln}\left(\left(\mathrm{1}+{k}\right)^{\mathrm{1}/{k}} \right)}\:=\frac{\mathrm{1}}{\mathrm{ln}\:{e}}=\frac{\mathrm{1}}{\mathrm{1}}=\mathrm{1} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\left({x}+{h}\right)−\mathrm{sin}\:{x}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{x}\:\mathrm{cos}\:{h}+\mathrm{cos}\:{x}\:\mathrm{sin}\:{h}−\mathrm{sin}\:{x}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{sin}\:{x}\frac{\mathrm{cos}\:{h}−\mathrm{1}}{{h}}+\mathrm{cos}\:{x}\frac{\mathrm{sin}\:{h}}{{h}}\right) \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{h}}{{h}}=\mathrm{1} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{cos}\:{h}−\mathrm{1}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{−\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \left({h}/\mathrm{2}\right)}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}−\frac{\mathrm{sin}^{\mathrm{2}} \left({h}/\mathrm{2}\right)}{\left({h}/\mathrm{2}\right)^{\mathrm{2}} }\centerdot\frac{\left({h}/\mathrm{2}\right)^{\mathrm{2}} }{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}−\left(\frac{\mathrm{sin}\left({h}/\mathrm{2}\right)}{{h}/\mathrm{2}}\right)^{\mathrm{2}} \centerdot\frac{{h}}{\mathrm{4}}=−\mathrm{1}\centerdot\mathrm{0}=\mathrm{0} \\ $$$$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{sin}\:{x}\centerdot\mathrm{0}+\mathrm{cos}\:{x}\centerdot\mathrm{1}\right)=\mathrm{cos}\:{x} \\ $$$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left({a}+{i}\frac{{b}−{a}}{{n}}\right)\frac{{b}−{a}}{{n}} \\ $$$$\int_{{a}} ^{{b}} {x}^{{m}} {dx}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left({k}\frac{{b}}{{n}}\right)^{{m}} \frac{{b}}{{n}}={b}^{{m}+\mathrm{1}} \underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\frac{{k}}{{n}}\right)^{{m}} \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\frac{{k}}{{n}}\right)^{{m}} \frac{{b}}{{n}}=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\frac{{n}^{{m}+\mathrm{1}} }{{m}+\mathrm{1}}+{O}\left({n}^{{m}} \right)}{{n}^{{m}+\mathrm{1}} }=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{m}+\mathrm{1}}+{O}\left(\frac{\mathrm{1}}{{n}}\right)\right)=\frac{\mathrm{1}}{{m}+\mathrm{1}} \\ $$$$\int_{{a}} ^{{b}} {x}^{{m}} {dx}={b}^{{m}+\mathrm{1}} \centerdot\frac{\mathrm{1}}{{m}+\mathrm{1}}=\frac{{b}^{{m}+\mathrm{1}} }{{m}+\mathrm{1}} \\ $$$$\int{x}^{{m}} {dx}=\frac{{x}^{{m}+\mathrm{1}} }{{m}+\mathrm{1}}+{C} \\ $$$${G}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} {f}\left({t}\right){dt} \\ $$$${G}'\left({x}\right)=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{G}\left({x}+{h}\right)−{G}\left({x}\right)}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\int_{{x}} ^{{x}+{h}} {f}\left({t}\right){dt}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{f}\left({c}_{{n}} \right){h}}{{h}}=\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}{f}\left({c}_{{n}} \right)={f}\left({x}\right)\:,{c}_{{h}} \in\left[{x},{x}+{h}\right] \\ $$$${F}\left({b}\right)−{F}\left({a}\right)=\left[\int_{{a}} ^{{b}} {f}\left({t}\right){dt}+{C}\right]−\left[\int_{{a}} ^{{a}} {f}\left({t}\right){dt}+{C}\right]=\int_{{a}} ^{{b}} {f}\left({t}\right){dt}−\mathrm{0}=\int_{{a}} ^{{b}} {f}\left({t}\right){dt} \\ $$

Question Number 221831    Answers: 1   Comments: 0

∫ ((√x)/( (√(1+x^2 )) + (√(1−x^2 )))) dx

$$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx} \\ $$$$ \\ $$

Question Number 221830    Answers: 0   Comments: 0

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

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\sqrt{{x}}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:+\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx} \\ $$$$ \\ $$

Question Number 221829    Answers: 4   Comments: 0

(√(70.71.72.73+1))

$$\sqrt{\mathrm{70}.\mathrm{71}.\mathrm{72}.\mathrm{73}+\mathrm{1}} \\ $$

Question Number 221815    Answers: 1   Comments: 0

Question Number 221814    Answers: 1   Comments: 0

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