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Question Number 219869 Answers: 1 Comments: 0

Question Number 219868 Answers: 1 Comments: 0

Prove that; (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = −cos 2x

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:−\mathrm{cos}\:\mathrm{2}{x}\:\:\:\: \\ $$$$ \\ $$

Question Number 219866 Answers: 1 Comments: 0

Prove that; ∫^( π/2) _( 0) sin^(2x−1) θ cos^(2y−1) θ dθ = (1/2) ((Γ(x)Γ(y))/(Γ(x)+Γ(y)))

$$\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\underset{\:\mathrm{0}} {\int}^{\:\pi/\mathrm{2}} \:\mathrm{sin}^{\mathrm{2}{x}−\mathrm{1}} \theta\:\mathrm{cos}\:^{\mathrm{2}{y}−\mathrm{1}} \theta\:{d}\theta\:=\:\frac{\mathrm{1}}{\mathrm{2}}\:\frac{\Gamma\left({x}\right)\Gamma\left({y}\right)}{\Gamma\left({x}\right)+\Gamma\left({y}\right)}\:\:\:\:\: \\ $$$$\: \\ $$

Question Number 219865 Answers: 1 Comments: 0

Prove that; ∫_( 0) ^( 1) lnΓ(x)dx = ln (√(2π))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{\:\mathrm{0}} {\overset{\:\mathrm{1}} {\int}}\:{ln}\Gamma\left({x}\right){dx}\:=\:{ln}\:\sqrt{\mathrm{2}\pi} \\ $$$$ \\ $$

Question Number 219864 Answers: 1 Comments: 0

Prove that; ∫_( 0) ^( 1) (x^( n+1) /(x+1)) dx < (1/(2(n+1)))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{x}^{\:{n}+\mathrm{1}} }{{x}+\mathrm{1}}\:{dx}\:<\:\frac{\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)} \\ $$$$ \\ $$

Question Number 219863 Answers: 1 Comments: 0

Prove that; ∫^( ∞) _( 0) ((ln x)/(x^3 + x(√x) + 1)) dx = −((32π)/(81))sin(π/(18))

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\underset{\:\mathrm{0}} {\int}^{\:\infty} \:\frac{{ln}\:{x}}{{x}^{\mathrm{3}} +\:{x}\sqrt{{x}}\:+\:\mathrm{1}}\:{dx}\:=\:−\frac{\mathrm{32}\pi}{\mathrm{81}}{sin}\frac{\pi}{\mathrm{18}}\:\:\:\: \\ $$$$ \\ $$

Question Number 219853 Answers: 1 Comments: 0

If 0<a≤b Then prove that ∫_a ^( b) (sinx)^(2sin^2 x) ∙ (1−sin^2 x)^(1−sin^2 x) dx ≥ ((b−a)/2)

$$\mathrm{If}\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \left(\mathrm{sinx}\right)^{\mathrm{2}\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\centerdot\:\left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \mathrm{x}\right)^{\mathrm{1}−\boldsymbol{\mathrm{sin}}^{\mathrm{2}} \boldsymbol{\mathrm{x}}} \:\mathrm{dx}\:\geqslant\:\frac{\mathrm{b}−\mathrm{a}}{\mathrm{2}} \\ $$

Question Number 219849 Answers: 0 Comments: 0

Question Number 219846 Answers: 1 Comments: 0

If f:[a,b]→R 0<a≤b f - continuous Then prove that ((b^(4047) − a^(4047) )/(4047)) + ∫_a ^( b) f^2 (x^(2024) ) dx ≥ (1/(1012)) ∫_a^(2024) ^( b^(2024) ) f(x) dx

$$\mathrm{If}\:\:\:\mathrm{f}:\left[\mathrm{a},\mathrm{b}\right]\rightarrow\mathbb{R} \\ $$$$\:\:\:\:\:\:\mathrm{0}<\mathrm{a}\leqslant\mathrm{b} \\ $$$$\mathrm{f}\:-\:\mathrm{continuous} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{b}^{\mathrm{4047}} \:−\:\mathrm{a}^{\mathrm{4047}} }{\mathrm{4047}}\:+\:\int_{\boldsymbol{\mathrm{a}}} ^{\:\boldsymbol{\mathrm{b}}} \:\mathrm{f}\:^{\mathrm{2}} \:\left(\mathrm{x}^{\mathrm{2024}} \right)\:\mathrm{dx}\:\geqslant\:\frac{\mathrm{1}}{\mathrm{1012}}\:\int_{\boldsymbol{\mathrm{a}}^{\mathrm{2024}} } ^{\:\boldsymbol{\mathrm{b}}^{\mathrm{2024}} } \:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$

Question Number 219844 Answers: 2 Comments: 0

Question Number 219843 Answers: 0 Comments: 0

If a,b,c,d > 0 a^2 +b^2 +c^2 +d^2 = 4 Then prove that (1/((1+ab)^3 )) + (1/((1+ac)^3 )) + (1/((1+ad)^3 )) + (1/((1+bc)^3 )) + (1/((1+bd)^3 )) + (1/((1+cd)^3 )) ≥ (3/4)

$$\mathrm{If}\:\:\:\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\:>\:\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\mathrm{a}^{\mathrm{2}} +\mathrm{b}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} +\mathrm{d}^{\mathrm{2}} \:=\:\mathrm{4} \\ $$$$\mathrm{Then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ab}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ac}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{ad}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bc}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{bd}\right)^{\mathrm{3}} }\:+\:\frac{\mathrm{1}}{\left(\mathrm{1}+\mathrm{cd}\right)^{\mathrm{3}} }\:\geqslant\:\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 219839 Answers: 1 Comments: 0