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IntegrationQuestion and Answers: Page 1

Question Number 227271 Answers: 3 Comments: 0

Question Number 227255 Answers: 8 Comments: 0

Question Number 227184 Answers: 3 Comments: 0

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Question Number 227146 Answers: 2 Comments: 0

Question Number 227128 Answers: 1 Comments: 0

Question Number 227127 Answers: 1 Comments: 2

prove: ∫_0 ^1 ((lnx)/(1+x^6 ))dx=(π^2 /(12(√3)))−(5/9)G

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{prove}: \\ $$$$\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}{x}}{\mathrm{1}+{x}^{\mathrm{6}} }{dx}=\frac{\pi^{\mathrm{2}} }{\mathrm{12}\sqrt{\mathrm{3}}}−\frac{\mathrm{5}}{\mathrm{9}}\boldsymbol{\mathrm{G}} \\ $$$$ \\ $$

Question Number 227113 Answers: 1 Comments: 0

∫_0 ^1 ⌊log_2 (x−2^(⌊log_2 x⌋) )⌋dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \lfloor\mathrm{log}_{\mathrm{2}} \left({x}−\mathrm{2}^{\lfloor\mathrm{log}_{\mathrm{2}} {x}\rfloor} \right)\rfloor{dx} \\ $$

Question Number 227108 Answers: 1 Comments: 0

∫_0 ^π ((sin 2x)/3)(√(cos x+1)) dx =?

$$\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{3}}\sqrt{\mathrm{cos}\:{x}+\mathrm{1}}\:{dx}\:=? \\ $$

Question Number 227055 Answers: 1 Comments: 0

A parabolic refector is formed by revolving the arc of the parabala y^2 =4ax from x=0 to x=h about the axis.If the diameter of the reflector is 2l.Show that the area of the reflecting surface is ((πl)/(6h^2 )){(l^2 +4h^2 )^(3/2) −l^3 }

$${A}\:{parabolic}\:{refector}\:{is}\:{formed}\:{by} \\ $$$${revolving}\:{the}\:{arc}\:{of}\:{the}\:{parabala} \\ $$$${y}^{\mathrm{2}} =\mathrm{4}{ax}\:\:{from}\:{x}=\mathrm{0}\:\:\:\:{to}\:\:{x}={h} \\ $$$${about}\:{the}\:{axis}.{If}\:{the}\:\:{diameter} \\ $$$${of}\:{the}\:{reflector}\:{is}\:\mathrm{2}{l}.{Show}\:{that} \\ $$$${the}\:{area}\:{of}\:{the}\:{reflecting}\:{surface}\:{is} \\ $$$$\frac{\pi{l}}{\mathrm{6}{h}^{\mathrm{2}} }\left\{\left({l}^{\mathrm{2}} +\mathrm{4}{h}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} −{l}^{\mathrm{3}} \right\} \\ $$$$ \\ $$

Question Number 226995 Answers: 3 Comments: 0

Question Number 226994 Answers: 4 Comments: 0

Question Number 226953 Answers: 1 Comments: 0

If I_n =∫(x^2 +a^2 )^n dx Show that I_n =(1/(2n+1))x(x^2 +a^2 )^n +2na^2 I_(n−1)

$${If}\:{I}_{{n}} =\int\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{{n}} {dx}\: \\ $$$${Show}\:{that} \\ $$$${I}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}{x}\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{{n}} +\mathrm{2}{na}^{\mathrm{2}} {I}_{{n}−\mathrm{1}} \: \\ $$

Question Number 226952 Answers: 4 Comments: 0

Question Number 226819 Answers: 1 Comments: 0

Evaluate ∫_0 ^∞ (dx/(1+x^2 ))

$${Evaluate} \\ $$$$\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$

Question Number 226818 Answers: 1 Comments: 0

Evaluate ∫((x^2 +2x−1)/(2x^3 +3x^2 −2x))dx

$${Evaluate} \\ $$$$\int\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}−\mathrm{1}}{\mathrm{2}{x}^{\mathrm{3}} +\mathrm{3}{x}^{\mathrm{2}} −\mathrm{2}{x}}{dx} \\ $$

Question Number 226799 Answers: 2 Comments: 0

∫_0 ^1 ((ln(1+x^2 ))/(1+x)) dx = ?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}\:\mathrm{d}{x}\:=\:? \\ $$

Question Number 226780 Answers: 1 Comments: 0

By using concept of complex number show that tan 5θ=((tan^5 θ−10tan^3 θ+5tan θ)/(5tan^4 θ−10tan^2 θ+1))

$${By}\:{using}\:{concept}\:{of}\:{complex} \\ $$$${number} \\ $$$${show}\:{that} \\ $$$$\mathrm{tan}\:\mathrm{5}\theta=\frac{\mathrm{tan}\:^{\mathrm{5}} \theta−\mathrm{10tan}\:^{\mathrm{3}} \theta+\mathrm{5tan}\:\theta}{\mathrm{5tan}\:^{\mathrm{4}} \theta−\mathrm{10tan}\:^{\mathrm{2}} \theta+\mathrm{1}} \\ $$

Question Number 226776 Answers: 4 Comments: 0

Approximate ∫_0 ^1 xe^x^2 dx with 6 ordinates. Use both rules Simpsons and Trapozoidal rules,hence evaluate and calculate the percentage error commetted for each case.Give comments

$${Approximate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {xe}^{{x}^{\mathrm{2}} } {dx}\:{with}\:\mathrm{6}\:{ordinates}. \\ $$$${Use}\:{both}\:{rules}\:{Simpsons}\:{and} \\ $$$${Trapozoidal}\:{rules},{hence}\:{evaluate}\:{and} \\ $$$${calculate}\:{the}\:{percentage}\:{error} \\ $$$${commetted}\:{for}\:{each}\:{case}.{Give}\:{comments} \\ $$$$ \\ $$

Question Number 226668 Answers: 1 Comments: 0

Question Number 226561 Answers: 4 Comments: 0

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Question Number 226339 Answers: 1 Comments: 0

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