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Question Number 196992 Answers: 0 Comments: 0

If A(a^2 , 2a), B((1/a^2 ), ((− 2)/a)) and S(1, 0) are three points then prove that, (1/(SA)) + (1/(SB)) = 1.

$$\mathrm{If}\:\mathrm{A}\left({a}^{\mathrm{2}} ,\:\mathrm{2}{a}\right),\:\mathrm{B}\left(\frac{\mathrm{1}}{{a}^{\mathrm{2}} },\:\frac{−\:\mathrm{2}}{{a}}\right)\:\mathrm{and}\:\mathrm{S}\left(\mathrm{1},\:\mathrm{0}\right)\: \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{points}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}, \\ $$$$\frac{\mathrm{1}}{\mathrm{SA}}\:+\:\frac{\mathrm{1}}{\mathrm{SB}}\:=\:\mathrm{1}. \\ $$

Question Number 196986 Answers: 1 Comments: 0

Prove that lim_(n→+∞) n∫^( 1) _( 0) e^(−nt(1−ln(1−t))) dt=(√(2π))

$$\mathrm{Prove}\:\mathrm{that}\: \\ $$$$\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}n}\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{e}^{−\mathrm{nt}\left(\mathrm{1}−\mathrm{ln}\left(\mathrm{1}−\mathrm{t}\right)\right)} \mathrm{dt}=\sqrt{\mathrm{2}\pi} \\ $$

Question Number 196983 Answers: 3 Comments: 1

Question Number 196976 Answers: 1 Comments: 0

Question Number 196973 Answers: 1 Comments: 0

Question Number 196972 Answers: 1 Comments: 0

Question Number 196971 Answers: 1 Comments: 0

solve { ((3x^2 −9y=1)),((3y^2 −9x=0)) :}

$$\:\:{solve}\:\begin{cases}{\mathrm{3}{x}^{\mathrm{2}} −\mathrm{9}{y}=\mathrm{1}}\\{\mathrm{3}{y}^{\mathrm{2}} −\mathrm{9}{x}=\mathrm{0}}\end{cases} \\ $$

Question Number 196968 Answers: 2 Comments: 0

Question Number 196965 Answers: 1 Comments: 0

Question Number 196964 Answers: 2 Comments: 0

Question Number 196959 Answers: 2 Comments: 0

lim_(x→0) (((sin 2x)/x^3 ) + (a/x^2 ) + b) = 1

$$\:\:\:\:\: \\ $$$$ \underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sin}\:\mathrm{2x}}{\mathrm{x}^{\mathrm{3}} }\:+\:\frac{\mathrm{a}}{\mathrm{x}^{\mathrm{2}} }\:+\:\mathrm{b}\right)\:=\:\mathrm{1} \\ $$

Question Number 196957 Answers: 1 Comments: 0

Question Number 196955 Answers: 0 Comments: 1

Question Number 196954 Answers: 1 Comments: 0

Question Number 196950 Answers: 1 Comments: 0

Prove that ∫^( (π/2)) _( 0) ((ln(1+αsint))/(sint))dt= (π^2 /8)−(1/2)(arccosα)^2

$$\mathrm{Prove}\:\mathrm{that}\:\underset{\:\mathrm{0}} {\int}^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{ln}\left(\mathrm{1}+\alpha\mathrm{sin}{t}\right)}{\mathrm{sin}{t}}{dt}=\:\frac{\pi^{\mathrm{2}} }{\mathrm{8}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{arccos}\alpha\right)^{\mathrm{2}} \\ $$

Question Number 196946 Answers: 2 Comments: 0

Question Number 196940 Answers: 1 Comments: 0

Question Number 196938 Answers: 1 Comments: 0

Question Number 196953 Answers: 1 Comments: 0

Question Number 196934 Answers: 1 Comments: 0

Question Number 196929 Answers: 1 Comments: 0

Question Number 196928 Answers: 0 Comments: 1

I_n =∫_0 ^(π/2) sin^n x dx

$${I}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{{n}} {x}\:{dx} \\ $$

Question Number 196918 Answers: 1 Comments: 0

Question Number 196917 Answers: 3 Comments: 0

Question Number 196916 Answers: 0 Comments: 0

Question Number 196915 Answers: 0 Comments: 1

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