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AllQuestion and Answers: Page 181

Question Number 201223 Answers: 1 Comments: 0

∫(√(1+(√(1+x)))) dx

$$\int\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\boldsymbol{{x}}}}\:\boldsymbol{{dx}} \\ $$

Question Number 201222 Answers: 1 Comments: 0

∫ (x^6 +x^9 )^(1/6) dx

$$\:\int\:\left(\boldsymbol{{x}}^{\mathrm{6}} +\boldsymbol{{x}}^{\mathrm{9}} \right)^{\frac{\mathrm{1}}{\mathrm{6}}} \boldsymbol{{dx}} \\ $$

Question Number 201221 Answers: 2 Comments: 0

Question Number 201214 Answers: 1 Comments: 0

A ball lies on the function z=xy at the point (1,2,2). Find the point in the xy−plane where the ball will touch it. (an unsolved old question Q200929)

$$\mathrm{A}\:\mathrm{ball}\:\mathrm{lies}\:\mathrm{on}\:\mathrm{the}\:\mathrm{function}\:{z}={xy}\:\mathrm{at} \\ $$$$\mathrm{the}\:\mathrm{point}\:\left(\mathrm{1},\mathrm{2},\mathrm{2}\right).\:\mathrm{Find}\:\mathrm{the}\:\mathrm{point}\:\mathrm{in} \\ $$$$\mathrm{the}\:{xy}−\mathrm{plane}\:\mathrm{where}\:\mathrm{the}\:\mathrm{ball}\:\mathrm{will} \\ $$$$\mathrm{touch}\:\mathrm{it}. \\ $$$$ \\ $$$$\left({an}\:{unsolved}\:{old}\:{question}\:{Q}\mathrm{200929}\right) \\ $$

Question Number 201209 Answers: 0 Comments: 4

Question Number 201200 Answers: 3 Comments: 2

Question Number 201192 Answers: 1 Comments: 0

Question Number 201190 Answers: 1 Comments: 0

Question Number 201185 Answers: 1 Comments: 0

Question Number 201172 Answers: 1 Comments: 0

Question Number 201166 Answers: 1 Comments: 0

Question Number 201162 Answers: 4 Comments: 0

Question Number 201152 Answers: 4 Comments: 0

if 4^x +4^(−x) =7 then 8^x +8^(−x) =?

$${if}\:\:\mathrm{4}^{{x}} +\mathrm{4}^{−{x}} =\mathrm{7} \\ $$$${then}\:\:\:\mathrm{8}^{{x}} +\mathrm{8}^{−{x}} =? \\ $$

Question Number 201150 Answers: 2 Comments: 0

Question Number 201149 Answers: 1 Comments: 0

Question Number 201146 Answers: 1 Comments: 4

Question Number 201144 Answers: 2 Comments: 0

(((14)/(15)))^6 ×(((45)/(28)))^6 =

$$\left(\frac{\mathrm{14}}{\mathrm{15}}\right)^{\mathrm{6}} ×\left(\frac{\mathrm{45}}{\mathrm{28}}\right)^{\mathrm{6}} = \\ $$

Question Number 201140 Answers: 1 Comments: 0

If R_− =x^2 yi_− −2y^2 zj_− +xy^2 z^2 k_− , find ∣(d^2 R/dx^2 )×(d^2 R/dy^2 )∣ at the point (2,1,−2)

$$\boldsymbol{{If}}\:\underset{−} {\boldsymbol{{R}}}=\boldsymbol{{x}}^{\mathrm{2}} \boldsymbol{{y}}\underset{−} {\boldsymbol{{i}}}−\mathrm{2}\boldsymbol{{y}}^{\mathrm{2}} \boldsymbol{{z}}\underset{−} {\boldsymbol{{j}}}+\boldsymbol{{xy}}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} \underset{−} {\boldsymbol{{k}}},\:\boldsymbol{{find}}\:\mid\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{R}}}{\boldsymbol{{dx}}^{\mathrm{2}} }×\frac{\boldsymbol{{d}}^{\mathrm{2}} \boldsymbol{{R}}}{\boldsymbol{{dy}}^{\mathrm{2}} }\mid\:\: \\ $$$$\boldsymbol{{at}}\:\boldsymbol{{the}}\:\boldsymbol{{point}}\:\left(\mathrm{2},\mathrm{1},−\mathrm{2}\right) \\ $$

Question Number 201139 Answers: 1 Comments: 1

Question Number 201135 Answers: 1 Comments: 0

Question Number 201134 Answers: 0 Comments: 0

S : Area of AB^Δ C in AB^Δ C : (a^( 2) /(4S)) =^? (1/2) (cot(B)+cot(C)) (a^( 2) /(4S)) = (( a^( 2) )/(4 ((1/2) bc sin(A))))=((4R^2 sin^( 2) (A))/(8R^( 2) sin(B)sin(C)sin(A))) = ((sin (A ))/(2sin(B)sin(C))) =^(A+B+C=π) ((sin(B)cos(C)+cosBsin(C))/(2sin(B)sin(C))) = (1/2) (cot(B)+cot(C)) ■

$$\: \\ $$$$\:\:\:\:\:\:{S}\::\:\:{Area}\:\:{of}\:\:\:{A}\overset{\Delta} {{B}C} \\ $$$$\:\:\:\:\:{in}\:\:\:\:{A}\overset{\Delta} {{B}C}\:\::\:\:\:\frac{{a}^{\:\mathrm{2}} }{\mathrm{4}{S}}\:\overset{?} {=}\:\frac{\mathrm{1}}{\mathrm{2}}\:\left({cot}\left({B}\right)+{cot}\left({C}\right)\right) \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\frac{{a}^{\:\mathrm{2}} }{\mathrm{4}{S}}\:=\:\frac{\:{a}^{\:\mathrm{2}} }{\mathrm{4}\:\left(\frac{\mathrm{1}}{\mathrm{2}}\:{bc}\:{sin}\left({A}\right)\right)}=\frac{\mathrm{4}{R}^{\mathrm{2}} {sin}^{\:\mathrm{2}} \left({A}\right)}{\mathrm{8}{R}^{\:\mathrm{2}} {sin}\left({B}\right){sin}\left({C}\right){sin}\left({A}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{{sin}\:\left({A}\:\right)}{\mathrm{2}{sin}\left({B}\right){sin}\left({C}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\overset{{A}+{B}+{C}=\pi} {=}\:\:\frac{{sin}\left({B}\right){cos}\left({C}\right)+{cosBsin}\left({C}\right)}{\mathrm{2}{sin}\left({B}\right){sin}\left({C}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\left({cot}\left({B}\right)+{cot}\left({C}\right)\right)\:\:\:\blacksquare \\ $$$$ \\ $$

Question Number 201133 Answers: 0 Comments: 1

Ω= ∫_1 ^( 3) (( 1)/( (√((x−1 )^3 )) + (√((x+1 )^3 )))) dx= ?

$$ \\ $$$$ \\ $$$$\:\:\:\Omega=\:\int_{\mathrm{1}} ^{\:\mathrm{3}} \frac{\:\mathrm{1}}{\:\sqrt{\left({x}−\mathrm{1}\:\right)^{\mathrm{3}} }\:+\:\sqrt{\left({x}+\mathrm{1}\:\right)^{\mathrm{3}} }}\:{dx}=\:?\:\:\: \\ $$$$ \\ $$

Question Number 201112 Answers: 1 Comments: 0

Question Number 201110 Answers: 1 Comments: 0

∫(1/( (√((x−a)^3 ))+(√((x+a)^3 ))))dx

$$\int\frac{\mathrm{1}}{\:\sqrt{\left({x}−{a}\right)^{\mathrm{3}} }+\sqrt{\left({x}+{a}\right)^{\mathrm{3}} }}{dx} \\ $$

Question Number 201108 Answers: 0 Comments: 6

Question Number 201106 Answers: 1 Comments: 0

calculate ... Σ_(n=1) ^∞ (( ζ(2n ))/(2^( n) .n)) = ?

$$ \\ $$$$\:\:\:\:\:{calculate}\:... \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\zeta\left(\mathrm{2}{n}\:\right)}{\mathrm{2}^{\:{n}} .{n}}\:=\:? \\ $$$$ \\ $$

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