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AllQuestion and Answers: Page 1711

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 .

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

Question Number 36413 Answers: 0 Comments: 1

let I_n = ∫_0 ^1 x^n (√(3+x))dx 1)calculate lim_(n→+∞) I_n 2) calculate lim_(n→+∞) n I_n

$${let}\:\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{3}+{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{lim}_{{n}\rightarrow+\infty} {I}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{lim}_{{n}\rightarrow+\infty} \:{n}\:{I}_{{n}} \\ $$

Question Number 36412 Answers: 1 Comments: 1

calculate I_λ =∫_0 ^λ e^(−x) ln(1+e^x )dx

$${calculate}\:{I}_{\lambda} \:=\int_{\mathrm{0}} ^{\lambda} \:{e}^{−{x}} {ln}\left(\mathrm{1}+{e}^{{x}} \right){dx} \\ $$

Question Number 36411 Answers: 0 Comments: 0

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

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

Question Number 36410 Answers: 0 Comments: 1

find the value of I_n = ∫_0 ^1 x^n (√(1−x))dx

$${find}\:{the}\:{value}\:{of}\:{I}_{{n}} =\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$

Question Number 36406 Answers: 2 Comments: 1

find the values of I = ∫_0 ^π cos^4 dx and J = ∫_0 ^π sin^4 dx .

$${find}\:{the}\:{values}\:{of}\:\:{I}\:=\:\int_{\mathrm{0}} ^{\pi} {cos}^{\mathrm{4}} {dx}\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\pi} \:{sin}^{\mathrm{4}} {dx}\:. \\ $$

Question Number 36398 Answers: 1 Comments: 1

Consider triangle ABC.If 206 lines are drawn from A to BC how many triangles are formed?

$${Consider}\:{triangle}\:{ABC}.{If}\:\mathrm{206} \\ $$$${lines}\:{are}\:{drawn}\:{from}\:{A}\:{to}\:{BC}\:{how} \\ $$$${many}\:{triangles}\:{are}\:{formed}? \\ $$

Question Number 36397 Answers: 2 Comments: 0

find I = ∫_1 ^2 (dx/(x(√(x+1)) +(x+1)(√x)))

$${find}\:{I}\:\:=\:\int_{\mathrm{1}} ^{\mathrm{2}} \:\:\:\:\:\frac{{dx}}{{x}\sqrt{{x}+\mathrm{1}}\:\:+\left({x}+\mathrm{1}\right)\sqrt{{x}}} \\ $$$$ \\ $$

Question Number 36394 Answers: 0 Comments: 0

let f(x)=artanx find L(f(x)) L mean laplace trsnsform.

$${let}\:{f}\left({x}\right)={artanx}\:{find}\:\:{L}\left({f}\left({x}\right)\right) \\ $$$${L}\:{mean}\:{laplace}\:{trsnsform}. \\ $$

Question Number 36393 Answers: 0 Comments: 0

let f(x) = (2/(sinx)) ,2π periodic odd developp f at fourier serie .

$${let}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{sinx}}\:\:\:,\mathrm{2}\pi\:{periodic}\:{odd} \\ $$$${developp}\:{f}\:{at}\:{fourier}\:{serie}\:. \\ $$

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