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IntegrationQuestion and Answers: Page 151

Question Number 102159 Answers: 0 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −x+3)^2 ( 2x^2 +5)))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{3}\right)^{\mathrm{2}} \left(\:\mathrm{2x}^{\mathrm{2}} \:+\mathrm{5}\right)} \\ $$

Question Number 102138 Answers: 0 Comments: 0

∫_0 ^1 ∫_(√y) ^0 (√(x^3 +1+ax^2 +bx+c)) dx dy

$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\sqrt{{y}}} ^{\mathrm{0}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}+{ax}^{\mathrm{2}} +{bx}+{c}}\:\:{dx}\:{dy} \\ $$

Question Number 102128 Answers: 4 Comments: 0

∫((x+1)/((√x) +1))dx=?

$$\:\int\frac{{x}+\mathrm{1}}{\sqrt{{x}}\:+\mathrm{1}}{dx}=? \\ $$

Question Number 102121 Answers: 3 Comments: 1

∫_(−∞) ^∞ ((sin(x+(π/2)))/(1+x^2 )) dx By real analysis

$$\int_{−\infty} ^{\infty} \frac{{sin}\left({x}+\frac{\pi}{\mathrm{2}}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:{By}\:{real}\:{analysis} \\ $$

Question Number 102115 Answers: 3 Comments: 3

Γ(s)ζ(s)=∫_0 ^∞ (x^(s−1) /(e^x +1))dx (Prove that) And prove 1+2+3+4+5+6+7+....∞=−(1/(12))

$$\Gamma\left({s}\right)\zeta\left({s}\right)=\int_{\mathrm{0}} ^{\infty} \frac{{x}^{{s}−\mathrm{1}} }{{e}^{{x}} +\mathrm{1}}{dx}\:\:\left({Prove}\:{that}\right) \\ $$$${And}\:{prove}\:\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+....\infty=−\frac{\mathrm{1}}{\mathrm{12}} \\ $$$$ \\ $$

Question Number 102127 Answers: 1 Comments: 0

∫_0 ^2 (1−x^2 )^3 dx=? and write the furmollah

$$\int_{\mathrm{0}} ^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{3}} {dx}=? \\ $$$$\:{and}\:{write}\:{the}\:{furmollah} \\ $$

Question Number 102099 Answers: 1 Comments: 0

∫sinx ∙ cosx ∙cos2x ∙ cos4x dx=?

$$\int{sinx}\:\centerdot\:{cosx}\:\centerdot{cos}\mathrm{2}{x}\:\centerdot\:{cos}\mathrm{4}{x}\:{dx}=? \\ $$

Question Number 102097 Answers: 0 Comments: 2

∫((cos^2 x −cos^2 x)/(cosx−cosx))dx=?

$$\int\frac{{cos}^{\mathrm{2}} {x}\:−{cos}^{\mathrm{2}} {x}}{{cosx}−{cosx}}{dx}=? \\ $$

Question Number 102065 Answers: 1 Comments: 0

∫ ((x^2 dx)/((1−x)(√x))) ?

$$\int\:\frac{\mathrm{x}^{\mathrm{2}} \:\mathrm{dx}}{\left(\mathrm{1}−\mathrm{x}\right)\sqrt{\mathrm{x}}}\:? \\ $$

Question Number 102064 Answers: 1 Comments: 0

∫ ((sin 3x)/(cos 5x. cos 2x)) dx ?

$$\int\:\frac{\mathrm{sin}\:\mathrm{3x}}{\mathrm{cos}\:\mathrm{5x}.\:\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\:? \\ $$

Question Number 102043 Answers: 0 Comments: 1

∫e^(√(ax+b)) dx

$$\int{e}^{\sqrt{{ax}+{b}}} {dx} \\ $$

Question Number 102001 Answers: 2 Comments: 0

evaluate ∫cos^3 xsin^3 xdx.

$${evaluate}\:\int\mathrm{cos}\:^{\mathrm{3}} {x}\mathrm{sin}\:^{\mathrm{3}} {xdx}. \\ $$

Question Number 101985 Answers: 1 Comments: 0

∫ ((3x^5 − x^4 + 9x^3 − 12x^2 − 2x + 1)/((x^3 − 1)^2 )) dx

$$\int\:\frac{\mathrm{3x}^{\mathrm{5}} \:−\:\mathrm{x}^{\mathrm{4}} \:+\:\mathrm{9x}^{\mathrm{3}} \:−\:\mathrm{12x}^{\mathrm{2}} \:−\:\mathrm{2x}\:\:+\:\:\mathrm{1}}{\left(\mathrm{x}^{\mathrm{3}} \:−\:\mathrm{1}\right)^{\mathrm{2}} }\:\mathrm{dx} \\ $$

Question Number 102058 Answers: 1 Comments: 0

∫_0 ^1 ((sin(logx))/(logx))dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{sin}\left({logx}\right)}{{logx}}{dx} \\ $$

Question Number 102183 Answers: 4 Comments: 0

∫ ((cos θ)/(sin θ−cos θ)) dθ ?

$$\int\:\frac{\mathrm{cos}\:\theta}{\mathrm{sin}\:\theta−\mathrm{cos}\:\theta}\:{d}\theta\:? \\ $$

Question Number 101970 Answers: 1 Comments: 0

∫_0 ^∞ ((Cos(ax))/(x^2 +b^2 )) dx

$$\int_{\mathrm{0}} ^{\infty} \:\frac{{Cos}\left({ax}\right)}{{x}^{\mathrm{2}} +{b}^{\mathrm{2}} }\:{dx} \\ $$

Question Number 101959 Answers: 1 Comments: 0

Starting from y=(4/(√π))t^((3/2) ) ∫_0 ^∞ x^3 e^(−tx^2 ) dx find (π/8)=?

$$ \\ $$$$ \\ $$$${Starting}\:{from} \\ $$$$\:\:\:\:\:\:\:\:{y}=\frac{\mathrm{4}}{\sqrt{\pi}}{t}^{\frac{\mathrm{3}}{\mathrm{2}}\:} \int_{\mathrm{0}} ^{\infty} {x}^{\mathrm{3}} {e}^{−{tx}^{\mathrm{2}} } {dx} \\ $$$${find}\:\:\:\frac{\pi}{\mathrm{8}}=? \\ $$

Question Number 101940 Answers: 1 Comments: 0

lim_(n→∞ ) (1/n^2 )(ne^((−1)/n^2 ) +ne^((−4)/n^2 ) +.....∞)

$$\underset{{n}\rightarrow\infty\:} {\mathrm{lim}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }\left({ne}^{\frac{−\mathrm{1}}{{n}^{\mathrm{2}} }} +{ne}^{\frac{−\mathrm{4}}{{n}^{\mathrm{2}} }} +.....\infty\right) \\ $$

Question Number 102313 Answers: 2 Comments: 0

Question Number 101835 Answers: 3 Comments: 0

∫_0 ^∞ (1/(e^x +1)) dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\frac{\mathrm{1}}{{e}^{{x}} +\mathrm{1}}\:{dx}\: \\ $$

Question Number 101833 Answers: 3 Comments: 0

∫ _(−1)^1 (√((1+x)/(1−x))) dx ?

$$\int\:_{−\mathrm{1}} ^{\mathrm{1}} \sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}\:{dx}\:?\: \\ $$

Question Number 101816 Answers: 1 Comments: 0

∫_1 ^( e) (((tan^(−1) x)/x)+((log)/(x^2 +1)))dx

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

Question Number 101808 Answers: 2 Comments: 0

∫_(1/3) ^1 (((x−x^3 )^(1/3) )/x^4 )dx

$$\int_{\frac{\mathrm{1}}{\mathrm{3}}} ^{\mathrm{1}} \frac{\left(\mathrm{x}−\mathrm{x}^{\mathrm{3}} \right)^{\mathrm{1}/\mathrm{3}} }{\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 101793 Answers: 1 Comments: 0

∫_1 ^2 ((logu)/(((√(u−1)))((√(u−1))+1)))du

$$\int_{\mathrm{1}} ^{\mathrm{2}} \frac{{logu}}{\left(\sqrt{{u}−\mathrm{1}}\right)\left(\sqrt{{u}−\mathrm{1}}+\mathrm{1}\right)}{du} \\ $$

Question Number 101791 Answers: 2 Comments: 0

(1/n^(3 ) )lim_(n→∞) (ne^(−((1/n))^2 ) +2ne^(−((2/n))^2 ) +....∞)

$$\frac{\mathrm{1}}{{n}^{\mathrm{3}\:\:} }\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left({ne}^{−\left(\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2}} } +\mathrm{2}{ne}^{−\left(\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2}} } +....\infty\right) \\ $$

Question Number 101783 Answers: 2 Comments: 0

∫_( 0) ^( 1) ((ln(x^2 + 1))/(x + 1))

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

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