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IntegrationQuestion and Answers: Page 257

Question Number 51910 Answers: 0 Comments: 4

Question Number 51876 Answers: 1 Comments: 2

Question Number 51841 Answers: 0 Comments: 2

Question Number 51834 Answers: 0 Comments: 1

calculatef(a)= ∫_0 ^∞ ((ln(1+at^2 ))/(1+t^4 ))dt with a>0. 2)find the value of ∫_0 ^∞ ((ln(3+t^2 ))/(1+t^4 ))dt.

$${calculatef}\left({a}\right)=\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{ln}\left(\mathrm{1}+{at}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}\:\:{with}\:{a}>\mathrm{0}. \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{ln}\left(\mathrm{3}+{t}^{\mathrm{2}} \right)}{\mathrm{1}+{t}^{\mathrm{4}} }{dt}. \\ $$

Question Number 51825 Answers: 1 Comments: 3

find f(α)=∫_0 ^1 ln(1+e^(−α) x)dx with α≥0

$${find}\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{e}^{−\alpha} {x}\right){dx}\:\:{with}\:\alpha\geqslant\mathrm{0} \\ $$

Question Number 51824 Answers: 1 Comments: 0

find ∫ (dx/(cosx +cos(2x)+cos(3x)))

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

Question Number 51812 Answers: 0 Comments: 4

Question Number 51733 Answers: 2 Comments: 3

Question Number 51680 Answers: 1 Comments: 2

Solve the equation: (1) z^3 + 1 − 10i = 0 (2) z^4 − i + 2 = 0

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}:\:\:\:\:\: \\ $$$$\:\left(\mathrm{1}\right)\:\:\:\:\:\mathrm{z}^{\mathrm{3}} \:+\:\mathrm{1}\:−\:\mathrm{10i}\:\:=\:\:\mathrm{0} \\ $$$$\:\left(\mathrm{2}\right)\:\:\:\:\:\mathrm{z}^{\mathrm{4}} \:−\:\mathrm{i}\:+\:\mathrm{2}\:\:=\:\:\mathrm{0} \\ $$

Question Number 51594 Answers: 0 Comments: 0

Question Number 51552 Answers: 0 Comments: 1

calculate ∫_0 ^(+∞) ((arctan(1+x^2 ))/(x^2 +4))dx

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\frac{{arctan}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 51551 Answers: 2 Comments: 2

find f(λ) = ∫_0 ^(π/4) (√(1+λtant))dt with λ>0 also calculate ∫_0 ^(π/4) ((tant)/(√(1+λtant)))dt.

$${find}\:{f}\left(\lambda\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \sqrt{\mathrm{1}+\lambda{tant}}{dt}\:\:\:{with}\:\lambda>\mathrm{0} \\ $$$${also}\:{calculate}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{tant}}{\sqrt{\mathrm{1}+\lambda{tant}}}{dt}. \\ $$

Question Number 51550 Answers: 1 Comments: 2

find ∫_0 ^1 (√(1+x^4 ))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$

Question Number 51526 Answers: 1 Comments: 1

Question Number 51510 Answers: 1 Comments: 0

Question Number 51501 Answers: 1 Comments: 0

Question Number 51456 Answers: 1 Comments: 0

∫ (x^8 /(x^6 + 64)) dx

$$\int\:\:\:\frac{\mathrm{x}^{\mathrm{8}} }{\mathrm{x}^{\mathrm{6}} \:+\:\mathrm{64}}\:\mathrm{dx} \\ $$

Question Number 51421 Answers: 2 Comments: 3

∫ ((tan^(−1) x)/x^2 ) dx

$$\int\:\:\frac{\mathrm{tan}^{−\mathrm{1}} \mathrm{x}}{\mathrm{x}^{\mathrm{2}} }\:\:\mathrm{dx} \\ $$

Question Number 51419 Answers: 1 Comments: 0

∫ ((√x)/(1 + (x)^(1/3) )) dx

$$\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{1}\:+\:\:\sqrt[{\mathrm{3}}]{\mathrm{x}}}\:\mathrm{dx} \\ $$

Question Number 51394 Answers: 1 Comments: 0

∫ (1/(1 + (√(tan x)))) dx

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

Question Number 51353 Answers: 4 Comments: 1

Question Number 51324 Answers: 2 Comments: 0

calculate ∫_0 ^(+∞) (dx/(1+x^(2 ) +x^4 ))

$${calculate}\:\int_{\mathrm{0}} ^{+\infty} \:\:\:\:\:\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}\:} \:+{x}^{\mathrm{4}} } \\ $$

Question Number 51215 Answers: 1 Comments: 0

∫_0 ^π e^((1+i)x) dx=...

$$\int_{\mathrm{0}} ^{\pi} {e}^{\left(\mathrm{1}+{i}\right){x}} {dx}=... \\ $$

Question Number 51188 Answers: 0 Comments: 0

find ∫ ((cos^2 x)/(cosx +2sinx))dx

$${find}\:\:\int\:\:\:\:\:\frac{{cos}^{\mathrm{2}} {x}}{{cosx}\:+\mathrm{2}{sinx}}{dx} \\ $$

Question Number 51186 Answers: 1 Comments: 1

calculate ∫_0 ^1 (([nx])/(2x+1))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\left[{nx}\right]}{\mathrm{2}{x}+\mathrm{1}}{dx} \\ $$

Question Number 51122 Answers: 1 Comments: 1

∫ (dx/(tgx−(√(tgx))))=?

$$\int\:\:\:\:\frac{\boldsymbol{\mathrm{dx}}}{\boldsymbol{\mathrm{tgx}}−\sqrt{\boldsymbol{\mathrm{tgx}}}}=? \\ $$

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