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Question Number 226953    Answers: 0   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: 0   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

Question Number 226554    Answers: 3   Comments: 0

Question Number 226339    Answers: 1   Comments: 0

Question Number 226292    Answers: 2   Comments: 0

Question Number 226290    Answers: 1   Comments: 0

Question Number 226178    Answers: 1   Comments: 0

Question Number 226148    Answers: 0   Comments: 0

Question Number 226147    Answers: 1   Comments: 0

Question Number 226113    Answers: 0   Comments: 0

Question Number 226003    Answers: 1   Comments: 1

If r^2 +r((√3)−(1/( (√3))))sin θ=(2/3) find A=∫_(π/6) ^( π/2) ((r^2 /2))dθ

$${If}\:\:{r}^{\mathrm{2}} +{r}\left(\sqrt{\mathrm{3}}−\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\right)\mathrm{sin}\:\theta=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$${find}\:{A}=\int_{\pi/\mathrm{6}} ^{\:\pi/\mathrm{2}} \left(\frac{{r}^{\mathrm{2}} }{\mathrm{2}}\right){d}\theta \\ $$$$\: \\ $$

Question Number 225994    Answers: 2   Comments: 0

Question Number 225788    Answers: 1   Comments: 25

∫_( (√2)−1) ^( y) (√((2(√2))y−1))dy

$$\int_{\:\sqrt{\mathrm{2}}−\mathrm{1}} ^{\:{y}} \sqrt{\left(\mathrm{2}\sqrt{\mathrm{2}}\right){y}−\mathrm{1}}{dy} \\ $$

Question Number 225629    Answers: 2   Comments: 0

Question Number 225391    Answers: 0   Comments: 0

Σ_(n = 1) ^∞ (− 1)^(n − 1) ((H_n H_(2n) ^((2)) )/n) = ?

$$\:\:\:\:\:\:\:\underset{\mathrm{n}\:\:\:=\:\:\:\mathrm{1}} {\overset{\infty} {\sum}}\:\left(−\:\:\:\:\mathrm{1}\right)^{\mathrm{n}\:\:\:\:−\:\:\:\mathrm{1}} \:\frac{\mathrm{H}_{\mathrm{n}} \:\mathrm{H}_{\mathrm{2n}} ^{\left(\mathrm{2}\right)} }{\mathrm{n}}\:\:\:\:\:\:\:=\:\:\:\:\:? \\ $$

Question Number 225382    Answers: 1   Comments: 0

Been a while guys ∫_0 ^( 1) ((xln(1+x))/(1+x^2 ))dx

$$\mathrm{Been}\:\mathrm{a}\:\mathrm{while}\:\mathrm{guys} \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{xln}\left(\mathrm{1}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 224920    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((x ln(1 + x) Li_2 (x))/(1 + x^2 )) dx

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

Question Number 224839    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((x tan^(− 1) (x) ln(1 − x))/(1 + x^2 )) dx

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

Question Number 224798    Answers: 0   Comments: 0

∫_( 0) ^( 1) ((arctan^2 (x) ln(1 − x))/x^2 ) dx

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

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