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IntegrationQuestion and Answers: Page 213

Question Number 73335    Answers: 1   Comments: 2

eplcit f(x)=∫_0 ^1 ln(x+t+t^2 )dt with x>(1/4) 2)calculate ∫_0 ^1 ln(t^2 +t +(√2))dt

$${eplcit}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({x}+{t}+{t}^{\mathrm{2}} \right){dt}\:\:\:\:\:\:{with}\:{x}>\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left({t}^{\mathrm{2}} \:+{t}\:+\sqrt{\mathrm{2}}\right){dt} \\ $$

Question Number 73333    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((cos(π +2x^2 ))/((x^2 +4)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{cos}\left(\pi\:+\mathrm{2}{x}^{\mathrm{2}} \right)}{\left({x}^{\mathrm{2}} \:+\mathrm{4}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 73331    Answers: 0   Comments: 1

calculate ∫_0 ^∞ ((ln(1+e^(−3x^2 ) ))/(3+x^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{1}+{e}^{−\mathrm{3}{x}^{\mathrm{2}} } \right)}{\mathrm{3}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 73293    Answers: 2   Comments: 2

Explicit f(x)= ∫_1 ^∞ ((lnt)/((x^2 +t^2 )^2 )) dt

$${Explicit}\:\:{f}\left({x}\right)=\:\int_{\mathrm{1}} ^{\infty} \:\frac{{lnt}}{\left({x}^{\mathrm{2}} +{t}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dt}\: \\ $$

Question Number 73279    Answers: 1   Comments: 0

Question Number 73275    Answers: 0   Comments: 2

∫(4/(x^2 (√(4−xδϰ)))) ?

$$ \\ $$$$ \\ $$$$\int\frac{\mathrm{4}}{{x}^{\mathrm{2}} \sqrt{\mathrm{4}−{x}\delta\varkappa}}\:\:\:\:? \\ $$$$ \\ $$

Question Number 73238    Answers: 1   Comments: 1

let 0<a<1 calculate ∫_0 ^∞ ((ln(t)t^(a−1) )/(1+t))dt and ∫_0 ^∞ ((ln^2 (t)t^(a−1) )/(1+t))dt

$${let}\:\mathrm{0}<{a}<\mathrm{1}\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}^{\mathrm{2}} \left({t}\right){t}^{{a}−\mathrm{1}} }{\mathrm{1}+{t}}{dt} \\ $$

Question Number 73231    Answers: 1   Comments: 1

find the sum of Σ_(n=0) ^∞ (n^2 −3n+1)e^(−n)

$${find}\:{the}\:{sum}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \left({n}^{\mathrm{2}} −\mathrm{3}{n}+\mathrm{1}\right){e}^{−{n}} \\ $$

Question Number 73230    Answers: 0   Comments: 0

calculate A_n =∫_0 ^∞ ((1+x^n )/(2+x^(2n) ))dx and J_n =∫_0 ^∞ ((2+x^(3n) )/(5+x^(7n) ))dx with n integr natural not 0

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}+{x}^{{n}} }{\mathrm{2}+{x}^{\mathrm{2}{n}} }{dx}\:\:{and}\:{J}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{2}+{x}^{\mathrm{3}{n}} }{\mathrm{5}+{x}^{\mathrm{7}{n}} }{dx} \\ $$$${with}\:{n}\:{integr}\:{natural}\:{not}\:\mathrm{0} \\ $$

Question Number 73225    Answers: 0   Comments: 0

calculate f(x)=∫_0 ^π ln(x^2 −2xcosθ +1)dθ with x real.

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} {ln}\left({x}^{\mathrm{2}} −\mathrm{2}{xcos}\theta\:+\mathrm{1}\right){d}\theta\:\:{with}\:{x}\:{real}. \\ $$

Question Number 73202    Answers: 2   Comments: 3

∫((2x^2 −1+2x(√(x^2 −1)))/(x^2 −x+(x−1)(√(x^2 −1))))dx=? ∫(dx/(x(√(x+1))(√((1−x)^3 ))))=?

$$\int\frac{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{1}+\mathrm{2}{x}\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{{x}^{\mathrm{2}} −{x}+\left({x}−\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} −\mathrm{1}}}{dx}=? \\ $$$$\int\frac{{dx}}{{x}\sqrt{{x}+\mathrm{1}}\sqrt{\left(\mathrm{1}−{x}\right)^{\mathrm{3}} }}=? \\ $$

Question Number 73258    Answers: 1   Comments: 0

Question Number 73182    Answers: 0   Comments: 3

calculate ∫_0 ^∞ xe^(−x^2 ) arctan(x−(1/x))dx

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

Question Number 73181    Answers: 1   Comments: 1

calculate ∫_1 ^(3 ) ((x−2)/(√(x^2 +x+1)))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}\:} \:\frac{{x}−\mathrm{2}}{\sqrt{{x}^{\mathrm{2}} +{x}+\mathrm{1}}}{dx} \\ $$

Question Number 73180    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((lnx)/((x+1)^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{lnx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 73179    Answers: 1   Comments: 1

caoculate ∫_0 ^∞ ((arctan(x^2 −1))/(2x^2 +1))dx

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

Question Number 73178    Answers: 1   Comments: 0

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

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

Question Number 73155    Answers: 0   Comments: 0

reposting a former question... ∫(((x)^(1/5) −1)/((√x)+1))dx= [t=(x)^(1/(10)) → dx=10(x^9 )^(1/(10)) dx] =10∫((t^9 (t−1))/(t^4 −t^3 +t^2 −t+1))dt= =10∫(t^6 −t^4 −t)dt+10∫((t(t^2 −t+1))/(t^4 −t^3 +t^2 −t+1))dt= =((10)/7)t^7 −2t^5 −5t^2 +(5+(√5))∫(t/(t^2 −((1−(√5))/3)t+1))dt+(5−(√5))∫(t/(t^2 −((1+(√5))/2)t+1))dt= and it′s easy to solve these

$$\mathrm{reposting}\:\mathrm{a}\:\mathrm{former}\:\mathrm{question}... \\ $$$$\int\frac{\sqrt[{\mathrm{5}}]{{x}}−\mathrm{1}}{\sqrt{{x}}+\mathrm{1}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt[{\mathrm{10}}]{{x}}\:\rightarrow\:{dx}=\mathrm{10}\sqrt[{\mathrm{10}}]{{x}^{\mathrm{9}} }{dx}\right] \\ $$$$=\mathrm{10}\int\frac{{t}^{\mathrm{9}} \left({t}−\mathrm{1}\right)}{{t}^{\mathrm{4}} −{t}^{\mathrm{3}} +{t}^{\mathrm{2}} −{t}+\mathrm{1}}{dt}= \\ $$$$=\mathrm{10}\int\left({t}^{\mathrm{6}} −{t}^{\mathrm{4}} −{t}\right){dt}+\mathrm{10}\int\frac{{t}\left({t}^{\mathrm{2}} −{t}+\mathrm{1}\right)}{{t}^{\mathrm{4}} −{t}^{\mathrm{3}} +{t}^{\mathrm{2}} −{t}+\mathrm{1}}{dt}= \\ $$$$=\frac{\mathrm{10}}{\mathrm{7}}{t}^{\mathrm{7}} −\mathrm{2}{t}^{\mathrm{5}} −\mathrm{5}{t}^{\mathrm{2}} +\left(\mathrm{5}+\sqrt{\mathrm{5}}\right)\int\frac{{t}}{{t}^{\mathrm{2}} −\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{3}}{t}+\mathrm{1}}{dt}+\left(\mathrm{5}−\sqrt{\mathrm{5}}\right)\int\frac{{t}}{{t}^{\mathrm{2}} −\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}{t}+\mathrm{1}}{dt}= \\ $$$$\mathrm{and}\:\mathrm{it}'\mathrm{s}\:\mathrm{easy}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{these} \\ $$

Question Number 73147    Answers: 1   Comments: 1

∫((x−6)/(x^3 +1))dx

$$\int\frac{{x}−\mathrm{6}}{{x}^{\mathrm{3}} +\mathrm{1}}{dx} \\ $$$$ \\ $$

Question Number 73144    Answers: 1   Comments: 1

calculte ∫ ((x+(√(2+x^2 )))/(x+1−(√(2+x^2 ))))dx

$${calculte}\:\int\:\:\frac{{x}+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{{x}+\mathrm{1}−\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 73037    Answers: 1   Comments: 1

Question Number 72988    Answers: 1   Comments: 1

calculate f(x)=∫_0 ^∞ (e^(−xt^2 ) /(4+t^2 ))dt with x>0

$${calculate}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−{xt}^{\mathrm{2}} } }{\mathrm{4}+{t}^{\mathrm{2}} }{dt}\:\:\:{with}\:{x}>\mathrm{0} \\ $$

Question Number 72986    Answers: 0   Comments: 0

calculate ∫_0 ^∞ ((e^(−x^2 ) cosx)/((x^2 +x+1)^2 ))dx

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

Question Number 73012    Answers: 4   Comments: 0

∫(√(tanx)) dx

$$\int\sqrt{{tan}\mathrm{x}}\:{d}\mathrm{x} \\ $$

Question Number 72888    Answers: 1   Comments: 5

let f(x)=∫_(π/6) ^(π/4) ((tant)/(2+x cost))dt with x real 1)determine a explicit form for f(x) 2)determine also g(x)=∫_(π/6) ^(π/4) ((tant)/((2+xcost)^2 ))dx 3) find the value of ∫_(π/6) ^(π/4) ((tant)/((2+3cost)))dt and ∫_(π/6) ^(π/4) ((tant)/((2+3cost)^2 ))dt

$${let}\:{f}\left({x}\right)=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tant}}{\mathrm{2}+{x}\:{cost}}{dt}\:\:{with}\:{x}\:{real} \\ $$$$\left.\mathrm{1}\right){determine}\:{a}\:{explicit}\:{form}\:{for}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right){determine}\:{also}\:{g}\left({x}\right)=\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tant}}{\left(\mathrm{2}+{xcost}\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tant}}{\left(\mathrm{2}+\mathrm{3}{cost}\right)}{dt}\:{and}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{tant}}{\left(\mathrm{2}+\mathrm{3}{cost}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 72889    Answers: 1   Comments: 1

calculate Σ_(n=1) ^∞ (((−1)^n )/((2n+1)n^2 ))

$${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}+\mathrm{1}\right){n}^{\mathrm{2}} } \\ $$

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