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f(x) = ((e^(3x) +e^(−3x) )/2) 1) determine f^(−1) (x) 2) calculate ∫_0 ^1 x f(x)dx and ∫_0 ^1 f(x)dx 3) calculate ∫ f^(−1) (x)dx 4) calculate u_n = ∫_0 ^π f(x)cos(nx)dx and v_n = ∫_0 ^n f(x)sin(nx)dx find nature of Σ (v_n /u_n ) ∫_0 ^1 xf(x) dx =(1/2) ∫_0 ^1 x e^(3x) dx +(1/2) ∫_0 ^1 x e^(−3x) dx (by parts) =(1/2){ [(x/3)e^(3x) ]_0 ^1 −(1/3)∫_0 ^1 e^(3x) dx +[−(x/3)e^(−3x) ]_0 ^1 +(1/3)∫_0 ^1 e^(−3x) dx} =(1/2){(e^3 /3) −(1/9)(e^3 −1) −(e^(−3) /3) −(1/9)(e^(−3) −1)}

$${f}\left({x}\right)\:\:=\:\:\frac{{e}^{\mathrm{3}{x}} \:+{e}^{−\mathrm{3}{x}} }{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}\:{f}\left({x}\right){dx}\:\:\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\:\:\int\:\:{f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:{f}\left({x}\right){cos}\left({nx}\right){dx}\:{and}\:{v}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{f}\left({x}\right){sin}\left({nx}\right){dx} \\ $$$${find}\:{nature}\:{of}\:\Sigma\:\frac{{v}_{{n}} }{{u}_{{n}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{xf}\left({x}\right)\:{dx}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{\mathrm{3}{x}} {dx}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{−\mathrm{3}{x}} {dx}\:\:\:\left({by}\:{parts}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\left[\frac{{x}}{\mathrm{3}}{e}^{\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:−\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{\mathrm{3}{x}} {dx}\:\:+\left[−\frac{{x}}{\mathrm{3}}{e}^{−\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:+\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−\mathrm{3}{x}} {dx}\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{{e}^{\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{\mathrm{3}} −\mathrm{1}\right)\:−\frac{{e}^{−\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{−\mathrm{3}} −\mathrm{1}\right)\right\} \\ $$$$ \\ $$

Question Number 42695 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (x^4 /(x^(8 ) +16))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{8}\:} \:+\mathrm{16}}{dx} \\ $$

Question Number 42680 Answers: 0 Comments: 2

calculale A_n (α) = ∫_(−∞) ^(+∞) ((cos(αx^n ))/(1+x^2 )) dx with n integr natural.

$${calculale}\:\:{A}_{{n}} \left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}^{{n}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{with} \\ $$$${n}\:{integr}\:{natural}. \\ $$$$ \\ $$

Question Number 42679 Answers: 0 Comments: 2

calculate ∫_(π/4) ^(π/3) ((sinx)/(cosx +tanx))dx .

$${calculate}\:\:\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\:\:\frac{{sinx}}{{cosx}\:+{tanx}}{dx}\:. \\ $$

Question Number 42708 Answers: 0 Comments: 1

calculate f(α)=∫_0 ^∞ ((e^(−2x) −e^(−x) )/x^2 ) e^(−αx^2 ) dx with α>0 1) find the value of ∫_0 ^∞ ((e^(−2x) −e^(−x) )/x^2 ) e^(−2x^2 ) dx

$${calculate}\:\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\mathrm{2}{x}} −{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\alpha{x}^{\mathrm{2}} } {dx}\:\:{with}\:\alpha>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\mathrm{2}{x}} \:−{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\mathrm{2}{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 42631 Answers: 1 Comments: 7

let f(x)=2(√(x−1)) −2x 1) find D_f 2) study the variation of f(x) 3 ) calculate ∫_1 ^3 f(x)dx 4) determine f^(−1) (x) and calculate ∫_1 ^3 f^(−1) (x)dx 5) find the values of A = ∫_1 ^3 ((f(x))/(f^(−1) (x)dx)) and B = ((∫_1 ^3 f(x))/(∫_1 ^3 f^(−1) (x))) dx.

$${let}\:{f}\left({x}\right)=\mathrm{2}\sqrt{{x}−\mathrm{1}}\:−\mathrm{2}{x} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{variation}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\:\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{calculate}\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$$\left.\mathrm{5}\right)\:\:{find}\:{the}\:{values}\:{of}\:\:{A}\:=\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{f}\left({x}\right)}{{f}^{−\mathrm{1}} \left({x}\right){dx}}\:{and}\: \\ $$$${B}\:=\:\frac{\int_{\mathrm{1}} ^{\mathrm{3}} \:\:{f}\left({x}\right)}{\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}^{−\mathrm{1}} \left({x}\right)}\:{dx}. \\ $$

Question Number 42630 Answers: 0 Comments: 0

let f(x) = e^x −2(√(x−3)) 1) find f^(−1) (x) 2) find ∫ f^(−1) (t)dt

$${let}\:{f}\left({x}\right)\:=\:{e}^{{x}} −\mathrm{2}\sqrt{{x}−\mathrm{3}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:\int\:\:{f}^{−\mathrm{1}} \left({t}\right){dt}\: \\ $$

Question Number 42629 Answers: 0 Comments: 0

find A_n =∫_0 ^∞ ((sin(nx))/(sh(2nx)))dx with n natural integr not 0.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({nx}\right)}{{sh}\left(\mathrm{2}{nx}\right)}{dx}\:\:{with}\:{n}\:{natural}\:{integr} \\ $$$${not}\:\mathrm{0}. \\ $$

Question Number 42628 Answers: 0 Comments: 2

calculate I = ∫_(π/3) ^(π/2) ((cos(2x))/(sin(x)+cosx))dx and J =∫_(π/3) ^(π/2) ((sin(2x))/(sin(x) +cos(x)))dx

$${calculate}\:\:{I}\:\:=\:\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{cos}\left(\mathrm{2}{x}\right)}{{sin}\left({x}\right)+{cosx}}{dx}\:{and} \\ $$$${J}\:=\int_{\frac{\pi}{\mathrm{3}}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{{sin}\left({x}\right)\:+{cos}\left({x}\right)}{dx} \\ $$

Question Number 42622 Answers: 0 Comments: 1

find ∫ th(2x+1)dx

$${find}\:\int\:\:{th}\left(\mathrm{2}{x}+\mathrm{1}\right){dx}\: \\ $$

Question Number 42621 Answers: 0 Comments: 1

find ∫ th(x)dx

$${find}\:\int\:{th}\left({x}\right){dx}\: \\ $$

Question Number 42605 Answers: 0 Comments: 3

let f(x) = ∫_(−∞) ^(+∞) ((arctan (xt^2 ))/(1+2t^2 ))dt 1) find a explicite form of f(x) 2) calculate ∫_0 ^∞ ((arctan(t^2 ))/(1+2t^2 ))dt and ∫_0 ^∞ ((arctan(2t^2 ))/(1+2t^2 ))dt

$${let}\:{f}\left({x}\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{arctan}\:\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{explicite}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left({t}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} }{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{arctan}\left(\mathrm{2}{t}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{2}{t}^{\mathrm{2}} }{dt} \\ $$$$ \\ $$

Question Number 42603 Answers: 0 Comments: 2

let f(x) =∫_0 ^1 ln(1+ixt)dt calculate f^, (x) (x from R).

$${let}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{ixt}\right){dt}\:\:{calculate}\:{f}^{,} \left({x}\right)\:\:\:\:\left({x}\:{from}\:{R}\right). \\ $$

Question Number 42593 Answers: 1 Comments: 1

Question Number 42580 Answers: 1 Comments: 1

Question Number 42579 Answers: 1 Comments: 1

∫ sinx/(√(1+sinx))

$$\int\:{sinx}/\sqrt{\mathrm{1}+{sinx}} \\ $$

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