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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

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Question Number 42579 Answers: 1 Comments: 1

∫ sinx/(√(1+sinx))

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

Question Number 42559 Answers: 5 Comments: 0

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Question Number 42550 Answers: 1 Comments: 3

2^(1/x) =(√x) Find x

$$\mathrm{2}^{\frac{\mathrm{1}}{{x}}} =\sqrt{{x}} \\ $$$${Find}\:{x} \\ $$

Question Number 43535 Answers: 2 Comments: 2

1) find the value of ∫_(π/4) ^(π/3) (√(1+tanθ))dθ .

$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\sqrt{\mathrm{1}+{tan}\theta}{d}\theta\:. \\ $$

Question Number 42549 Answers: 1 Comments: 0

If y = a^(1/(1 − log_a x)) and z = a^(1/(1 − log_a y)) . show that x = a^(1/(1 − log_a z))

$$\mathrm{If}\:\:\:\mathrm{y}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{x}}} \:\:\:\:\:\mathrm{and}\:\:\:\:\:\:\:\:\mathrm{z}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{y}}} \:\:.\:\:\mathrm{show}\:\mathrm{that}\:\:\:\:\:\mathrm{x}\:=\:\mathrm{a}^{\frac{\mathrm{1}}{\mathrm{1}\:−\:\mathrm{log}_{\mathrm{a}} \mathrm{z}}} \\ $$

Question Number 42543 Answers: 0 Comments: 3

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let S_n =Σ_(k=1) ^n (e^(k/n) /n) find lim_(n→+∞) S_n

$${let}\:{S}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\:\frac{{e}^{\frac{{k}}{{n}}} }{{n}} \\ $$$${find}\:{lim}_{{n}\rightarrow+\infty} {S}_{{n}} \\ $$

Question Number 42521 Answers: 1 Comments: 0

Let a^(→ ) , b^→ , c^→ be three unit vectors such that 3a^→ +4b^→ +5c^→ = 0. Then prove that a^(→ ) , b^→ ,c^→ are coplanar.

$$\mathrm{Let}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}}\:,\:\overset{\rightarrow} {\mathrm{c}}\:\mathrm{be}\:\mathrm{three}\:\mathrm{unit}\:\mathrm{vectors} \\ $$$$\mathrm{such}\:\mathrm{that}\:\mathrm{3}\overset{\rightarrow} {\mathrm{a}}+\mathrm{4}\overset{\rightarrow} {\mathrm{b}}+\mathrm{5}\overset{\rightarrow} {\mathrm{c}}\:=\:\mathrm{0}.\:\mathrm{Then}\:\mathrm{prove} \\ $$$$\mathrm{that}\:\overset{\rightarrow\:} {\mathrm{a}},\:\overset{\rightarrow} {\mathrm{b}},\overset{\rightarrow} {\mathrm{c}}\:\mathrm{are}\:\mathrm{coplanar}. \\ $$

Question Number 42520 Answers: 1 Comments: 0

cos^3 A.sin3A+sinA.cos3A=(3/4)sin4A

$${cos}^{\mathrm{3}} \:{A}.{sin}\mathrm{3}{A}+{sinA}.{cos}\mathrm{3}{A}=\frac{\mathrm{3}}{\mathrm{4}}{sin}\mathrm{4}{A} \\ $$

Question Number 42519 Answers: 1 Comments: 0

cosec2A+cosec4A+cosec8A=cotA−cot8A(prlve ghis)

$${cosec}\mathrm{2}{A}+{cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}={cotA}−{cot}\mathrm{8}{A}\left({prlve}\:{ghis}\right) \\ $$

Question Number 42516 Answers: 1 Comments: 1

(x−(1/2)a),(x−(5/2)a)=? Solve Please.

$$\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{a}\right),\left(\mathrm{x}−\frac{\mathrm{5}}{\mathrm{2}}\mathrm{a}\right)=? \\ $$$$\mathrm{Solve}\:\mathrm{Please}. \\ $$

Question Number 42514 Answers: 1 Comments: 0

calculate d(x!)/dx= ?

$${calculate}\:\:{d}\left({x}!\right)/{dx}=\:? \\ $$

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