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Question Number 36440    Answers: 0   Comments: 0

find ∫ (dx/((3+x^2 )^(1/3) ))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} } \\ $$

Question Number 36439    Answers: 1   Comments: 1

calculate ∫_0 ^π ((sin(2x))/(2 +cosx))dx

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

Question Number 36438    Answers: 1   Comments: 3

let F(x) =∫_x ^(1/x) ((arctan(t))/t)dt 1) calculate (dF/dx)(x) 2) find F(x).

$${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{\frac{\mathrm{1}}{{x}}} \:\:\frac{{arctan}\left({t}\right)}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\frac{{dF}}{{dx}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{F}\left({x}\right). \\ $$

Question Number 36437    Answers: 0   Comments: 3

simplify Σ_(k=0) ^n (C_n ^k /(k+1))

$${simplify}\:\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\:\:\:\frac{{C}_{{n}} ^{{k}} }{{k}+\mathrm{1}} \\ $$

Question Number 36436    Answers: 2   Comments: 0

find ∫ ((sinx)/(1+cos^3 x))dx

$${find}\:\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+{cos}^{\mathrm{3}} {x}}{dx} \\ $$

Question Number 36435    Answers: 0   Comments: 4

find the value of h(t)=∫_0 ^1 ln(1+tx^2 ) with ∣t∣≤1 2) calculate ∫_0 ^1 ln(1+x^2 )dx 3) calculate ∫_0 ^1 ln(1−x^2 )dx

$${find}\:{the}\:{value}\:{of}\:{h}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)\:\:{with}\:\mid{t}\mid\leqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 36434    Answers: 1   Comments: 1

find g(x) =∫_0 ^x (e^(−t) /(√(1+t^2 )))dt.

$${find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\:\frac{{e}^{−{t}} }{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{dt}. \\ $$

Question Number 36433    Answers: 2   Comments: 1

valculate f(x)= ∫_0 ^2 (((x+2)^2 )/(√(x^2 +4x+5)))dx

$${valculate}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\frac{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{4}{x}+\mathrm{5}}}{dx} \\ $$

Question Number 36432    Answers: 1   Comments: 1

calculate I = ∫_(−∞) ^(+∞) ((x+1)/((x^2 +1)^2 ))dx .

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

Question Number 36431    Answers: 0   Comments: 1

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

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

Question Number 36430    Answers: 1   Comments: 1

find ∫ ((ln(x+x^2 ))/x^2 )dx

$${find}\:\:\int\:\:\:\frac{{ln}\left({x}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 36429    Answers: 1   Comments: 1

let ϕ(λ) = ∫_(λ/π) ^(π/λ) (1+(1/x^2 ))arctan(x)dx with λ>0 1) find a simple form of ϕ(λ) 2) calculate ϕ^′ (λ).

$${let}\:\:\varphi\left(\lambda\right)\:=\:\int_{\frac{\lambda}{\pi}} ^{\frac{\pi}{\lambda}} \left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctan}\left({x}\right){dx}\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:\varphi\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\varphi^{'} \left(\lambda\right). \\ $$

Question Number 36428    Answers: 2   Comments: 1

find ∫ (dx/(cos^4 x +sin^4 x))

$${find}\:\:\int\:\:\:\:\frac{{dx}}{{cos}^{\mathrm{4}} {x}\:+{sin}^{\mathrm{4}} {x}} \\ $$

Question Number 36427    Answers: 1   Comments: 1

calculate ∫_(π/8) ^(π/6) (dx/(sin(2x)))

$${calculate}\:\int_{\frac{\pi}{\mathrm{8}}} ^{\frac{\pi}{\mathrm{6}}} \:\:\:\frac{{dx}}{{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 36426    Answers: 1   Comments: 0

find ∫ x^2 (√(1+x^3 )) dx

$${find}\:\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{1}+{x}^{\mathrm{3}} \:}\:{dx} \\ $$

Question Number 36425    Answers: 2   Comments: 2

calculate ∫_1 ^4 ((x(√x))/(x^2 −5x +4))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} \:\:\:\frac{{x}\sqrt{{x}}}{{x}^{\mathrm{2}} \:−\mathrm{5}{x}\:+\mathrm{4}}{dx} \\ $$

Question Number 36424    Answers: 0   Comments: 2

find ∫ x^2 (√(x^2 −1))dx

$${find}\:\int\:\:{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{2}} −\mathrm{1}}{dx} \\ $$

Question Number 36423    Answers: 0   Comments: 0

find ∫ (dt/(t(√(t^2 +t+1))))

$${find}\:\int\:\:\:\:\:\frac{{dt}}{{t}\sqrt{{t}^{\mathrm{2}} \:+{t}+\mathrm{1}}} \\ $$

Question Number 36422    Answers: 0   Comments: 1

find f(x)= ∫_0 ^x (t^2 +1)arctan(t)dt .

$${find}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{{x}} \left({t}^{\mathrm{2}} +\mathrm{1}\right){arctan}\left({t}\right){dt}\:. \\ $$

Question Number 36421    Answers: 0   Comments: 1

find ∫ (dx/(1+2(√(1−x))))

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

Question Number 36420    Answers: 0   Comments: 0

let f(x)=∫_0 ^∞ (( arctan(xt^2 ))/(1+t^4 ))dt 1) calculate f^′ (x) 2) find a simple form of f(x).

$${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\:{arctan}\left({xt}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:{f}\left({x}\right). \\ $$

Question Number 36419    Answers: 0   Comments: 3

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

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

Question Number 36418    Answers: 1   Comments: 0

calculate ∫_(−3) ^4 ∣x^2 −2x−3∣dx

$${calculate}\:\:\int_{−\mathrm{3}} ^{\mathrm{4}} \mid{x}^{\mathrm{2}} \:−\mathrm{2}{x}−\mathrm{3}\mid{dx} \\ $$

Question Number 36417    Answers: 1   Comments: 1

calculate ∫_2 ^6 (dx/((√(x+1)) +(√(x−1))))

$${calculate}\:\int_{\mathrm{2}} ^{\mathrm{6}} \:\:\:\frac{{dx}}{\sqrt{{x}+\mathrm{1}}\:+\sqrt{{x}−\mathrm{1}}} \\ $$

Question Number 36416    Answers: 0   Comments: 0

calculate I = ∫_1 ^2 ((2x^3 +5x^2 −4x−7)/((x+2)^2 ))dx

$${calculate}\:{I}\:=\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\frac{\mathrm{2}{x}^{\mathrm{3}} \:+\mathrm{5}{x}^{\mathrm{2}} \:−\mathrm{4}{x}−\mathrm{7}}{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 36415    Answers: 0   Comments: 1

calculate I = ∫_0 ^(π/2) (x^3 +x)cos^2 xdx and J = ∫_0 ^(π/2) (x^3 +x)sin^2 xdx cslculate I and J .

$${calculate}\:{I}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left({x}^{\mathrm{3}} \:+{x}\right){cos}^{\mathrm{2}} {xdx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\left({x}^{\mathrm{3}} \:+{x}\right){sin}^{\mathrm{2}} {xdx} \\ $$$${cslculate}\:{I}\:{and}\:{J}\:. \\ $$

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