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Question Number 108638 Answers: 1 Comments: 0

((bemath)/★) prove that (((n−1)),(( r)) ) + (((n−1)),((r−1)) ) = ((n),(r) )

$$\:\:\:\frac{{bemath}}{\bigstar} \\ $$$${prove}\:{that}\:\begin{pmatrix}{{n}−\mathrm{1}}\\{\:\:\:\:\:{r}}\end{pmatrix}\:+\:\begin{pmatrix}{{n}−\mathrm{1}}\\{{r}−\mathrm{1}}\end{pmatrix}\:=\:\begin{pmatrix}{{n}}\\{{r}}\end{pmatrix} \\ $$

Question Number 108637 Answers: 2 Comments: 0

Question Number 108628 Answers: 0 Comments: 4

Prove that the inequality ∣cos x∣ ≥ 1 − sin^2 x hold true for all x ∈ R

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{inequality}\:\:\mid\mathrm{cos}\:\mathrm{x}\mid\:\:\geqslant\:\:\:\mathrm{1}\:\:−\:\:\mathrm{sin}^{\mathrm{2}} \mathrm{x}\:\:\:\:\mathrm{hold}\:\mathrm{true}\:\mathrm{for}\:\mathrm{all}\:\:\mathrm{x}\:\in\:\mathbb{R} \\ $$

Question Number 108626 Answers: 1 Comments: 0

Question Number 108623 Answers: 3 Comments: 0

((⊃BeMath⊃)/★) (1)lim_(b→a) ((b(√a)−a(√b))/(a(√a)+b(√a)−2a(√a))) (2) lim_(x→0) ((3e^(2x) +e^x −4)/x)

$$\:\:\frac{\supset\mathcal{B}{e}\mathcal{M}{ath}\supset}{\bigstar} \\ $$$$\:\left(\mathrm{1}\right)\underset{{b}\rightarrow{a}} {\mathrm{lim}}\:\frac{{b}\sqrt{{a}}−{a}\sqrt{{b}}}{{a}\sqrt{{a}}+{b}\sqrt{{a}}−\mathrm{2}{a}\sqrt{{a}}} \\ $$$$\left(\mathrm{2}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}{e}^{\mathrm{2}{x}} +{e}^{{x}} −\mathrm{4}}{{x}} \\ $$

Question Number 108619 Answers: 1 Comments: 3

Question Number 108614 Answers: 0 Comments: 0

Question Number 108612 Answers: 0 Comments: 0

Question Number 108609 Answers: 1 Comments: 1

1 + 2 + 3 + 4 + 5 + ... ∞ = ???

$$\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:\mathrm{5}\:+\:...\:\infty\:=\:??? \\ $$

Question Number 108605 Answers: 2 Comments: 0

((BeMath)/≊) ∫ (x^(11) /((x^8 +1)^2 )) dx

$$\:\:\:\frac{\boldsymbol{{B}}{e}\boldsymbol{{M}}{ath}}{\approxeq} \\ $$$$\:\int\:\frac{{x}^{\mathrm{11}} }{\left({x}^{\mathrm{8}} +\mathrm{1}\right)^{\mathrm{2}} }\:{dx}\: \\ $$

Question Number 108597 Answers: 3 Comments: 0

((∠ BeMath∠)/∦) (1) ∫ cos (ln x) dx (2) ∫ sin (ln x) dx

$$\:\:\:\:\:\frac{\angle\:\mathcal{B}{e}\mathcal{M}{ath}\angle}{\nparallel} \\ $$$$\left(\mathrm{1}\right)\:\int\:\mathrm{cos}\:\left(\mathrm{ln}\:{x}\right)\:{dx}\: \\ $$$$\left(\mathrm{2}\right)\:\int\:\mathrm{sin}\:\left(\mathrm{ln}\:{x}\right)\:{dx}\: \\ $$

Question Number 108594 Answers: 2 Comments: 0

Question Number 108593 Answers: 1 Comments: 0

Question Number 108588 Answers: 2 Comments: 1

Question Number 108584 Answers: 0 Comments: 0

find ∫_0 ^1 ((ln(1+x^2 ))/(x+2))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{x}+\mathrm{2}}\mathrm{dx} \\ $$

Question Number 108583 Answers: 1 Comments: 0

find ∫_0 ^1 ((ln(1+x^2 ))/(1+x^2 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 108582 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((ln(1+x))/(1+x^2 )) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{ln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 108573 Answers: 3 Comments: 0

please: in AB^Δ C prove that: ((cos(A))/(sin(B)sin(C))) +((cos(B))/(sin(A)sin(C)))+((cos(C))/(sin(A)sin(B))) =2

$$\:\:\:\:\:\:\:\:\:\mathrm{please}:\:\:\:\mathrm{in}\:\mathrm{A}\overset{\Delta} {\mathrm{B}C}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\:\frac{{cos}\left(\mathrm{A}\right)}{{sin}\left(\mathrm{B}\right){sin}\left(\mathrm{C}\right)}\:+\frac{{cos}\left(\mathrm{B}\right)}{{sin}\left(\mathrm{A}\right){sin}\left(\mathrm{C}\right)}+\frac{{cos}\left(\mathrm{C}\right)}{{sin}\left(\mathrm{A}\right){sin}\left(\mathrm{B}\right)}\:=\mathrm{2}\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 108571 Answers: 2 Comments: 0

Question Number 108570 Answers: 0 Comments: 0

Question Number 108568 Answers: 0 Comments: 0

Question Number 108553 Answers: 0 Comments: 0

If a_1 , a_2 , a_3 , be an AP, then prove that: Σ_(n = 1) ^(2m) (− 1)^(n − 1) a_n ^2 = (m/(2m − 1))(a_n ^2 − a_(2m) ^2 )

$$\mathrm{If}\:\:\:\:\mathrm{a}_{\mathrm{1}} ,\:\:\mathrm{a}_{\mathrm{2}} ,\:\:\mathrm{a}_{\mathrm{3}} ,\:\:\:\:\mathrm{be}\:\mathrm{an}\:\mathrm{AP},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\:\:\:\underset{\mathrm{n}\:\:=\:\:\mathrm{1}} {\overset{\mathrm{2m}} {\sum}}\:\left(−\:\mathrm{1}\right)^{\mathrm{n}\:\:−\:\:\mathrm{1}} \:\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:\:=\:\:\:\frac{\mathrm{m}}{\mathrm{2m}\:\:−\:\:\mathrm{1}}\left(\mathrm{a}_{\mathrm{n}} ^{\mathrm{2}} \:\:\:−\:\:\mathrm{a}_{\mathrm{2m}} ^{\mathrm{2}} \right) \\ $$

Question Number 108538 Answers: 0 Comments: 1

Question Number 108548 Answers: 0 Comments: 0

Question Number 108547 Answers: 1 Comments: 0

Solve the following equation: cosz =2

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{following}\:\mathrm{equation}: \\ $$$$\mathrm{cosz}\:=\mathrm{2} \\ $$

Question Number 108507 Answers: 1 Comments: 0

∫_0 ^(π/6) (√((3cos2x−1)/(cos^2 (x)))) dx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \sqrt{\frac{\mathrm{3}{cos}\mathrm{2}{x}−\mathrm{1}}{{cos}^{\mathrm{2}} \left({x}\right)}}\:{dx} \\ $$

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