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AllQuestion and Answers: Page 1630

Question Number 44179 Answers: 1 Comments: 1

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

$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}} \\ $$$$\left.\mathrm{2}\right){calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\:\frac{{dx}}{\sqrt{{x}^{\mathrm{2}} \:+{x}+\mathrm{1}}+\sqrt{{x}^{\mathrm{2}} −{x}\:+\mathrm{1}}} \\ $$

Question Number 44178 Answers: 0 Comments: 0

find f(x)=∫_0 ^π ((sin^2 t)/((x^2 −2x cost +1)^2 ))dt 2)find the value of ∫_0 ^π ((sin^2 t)/((x^2 −cost +1)^2 ))dt

$${find}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −\mathrm{2}{x}\:{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{sin}^{\mathrm{2}} {t}}{\left({x}^{\mathrm{2}} −{cost}\:+\mathrm{1}\right)^{\mathrm{2}} }{dt} \\ $$

Question Number 44176 Answers: 0 Comments: 1

find A_n =∫_0 ^∞ (t^n −[t])e^(−nt) dt and lim_(n→+∞) A_n n integr natural.

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \left({t}^{{n}} −\left[{t}\right]\right){e}^{−{nt}} {dt} \\ $$$${and}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} \\ $$$${n}\:{integr}\:{natural}. \\ $$

Question Number 44175 Answers: 0 Comments: 2

find f(a) =∫_1 ^(+∞) (dx/(ch^2 x +a sh^2 x)) 2) find the value of ∫_1 ^(+∞) (dx/(ch^2 x+2sh^2 x))

$${find}\:\:{f}\left({a}\right)\:=\int_{\mathrm{1}} ^{+\infty} \:\:\frac{{dx}}{{ch}^{\mathrm{2}} {x}\:+{a}\:{sh}^{\mathrm{2}} {x}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{1}} ^{+\infty} \:\:\:\frac{{dx}}{{ch}^{\mathrm{2}} {x}+\mathrm{2}{sh}^{\mathrm{2}} {x}} \\ $$

Question Number 44174 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) (dx/((1+x^2 )(1+x e^(iθ) )))

$${find}\:\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}\:{e}^{{i}\theta} \right)} \\ $$

Question Number 44173 Answers: 1 Comments: 1

calculate ∫_0 ^∞ (dt/((3+t^2 )(√(1+t))))dt

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dt}}{\left(\mathrm{3}+{t}^{\mathrm{2}} \right)\sqrt{\mathrm{1}+{t}}}{dt} \\ $$

Question Number 44161 Answers: 2 Comments: 0

LOL! found this on the web: 1=(√1)=(√((−1)(−1)))=(√(−1))(√(−1))=i^2 =−1 each step seems right, so where′s the mistake?

$$\mathrm{LOL}!\:\mathrm{found}\:\mathrm{this}\:\mathrm{on}\:\mathrm{the}\:\mathrm{web}: \\ $$$$\mathrm{1}=\sqrt{\mathrm{1}}=\sqrt{\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)}=\sqrt{−\mathrm{1}}\sqrt{−\mathrm{1}}=\mathrm{i}^{\mathrm{2}} =−\mathrm{1} \\ $$$$\mathrm{each}\:\mathrm{step}\:\mathrm{seems}\:\mathrm{right},\:\mathrm{so}\:\mathrm{where}'\mathrm{s}\:\mathrm{the}\:\mathrm{mistake}? \\ $$

Question Number 44159 Answers: 3 Comments: 0

If sin^(−1) x + sin^(−1) y + sin^(−1) z =π prove that : a) x(√(1−x^2 )) + y(√(1−y^2 )) +z(√(1−z^2 ))= 2xyz b) x^4 +y^4 +z^4 +4x^2 y^2 z^2 = 2(x^2 y^2 +y^2 z^2 +z^2 x^2 ).

$${If}\:\mathrm{sin}^{−\mathrm{1}} {x}\:+\:\mathrm{sin}^{−\mathrm{1}} {y}\:+\:\mathrm{sin}^{−\mathrm{1}} \boldsymbol{{z}}\:=\pi\: \\ $$$${prove}\:{that}\:: \\ $$$$\left.{a}\right)\:{x}\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }\:+\:{y}\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }\:+\boldsymbol{{z}}\sqrt{\mathrm{1}−\boldsymbol{{z}}^{\mathrm{2}} }=\:\mathrm{2}{xy}\boldsymbol{{z}} \\ $$$$\left.{b}\right)\:{x}^{\mathrm{4}} +{y}^{\mathrm{4}} +\boldsymbol{{z}}^{\mathrm{4}} +\mathrm{4}{x}^{\mathrm{2}} {y}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} =\:\mathrm{2}\left({x}^{\mathrm{2}} {y}^{\mathrm{2}} +{y}^{\mathrm{2}} \boldsymbol{{z}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \boldsymbol{{x}}^{\mathrm{2}} \right). \\ $$

Question Number 44148 Answers: 1 Comments: 0

∫dx/sinx∙sin(x+α)=?

$$\int{dx}/{sinx}\centerdot{sin}\left({x}+\alpha\right)=? \\ $$

Question Number 44142 Answers: 1 Comments: 0

Find the total number of integral sided triangle whose largest side is 30. please explain with workings

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{integral}\:\mathrm{sided}\:\mathrm{triangle}\:\mathrm{whose}\:\mathrm{largest}\:\mathrm{side} \\ $$$$\mathrm{is}\:\mathrm{30}. \\ $$$$\mathrm{please}\:\mathrm{explain}\:\mathrm{with}\:\mathrm{workings} \\ $$

Question Number 44128 Answers: 1 Comments: 0

The number of all possible 5−tuples (a_1 , a_2 , a_3 , a_4 , a_5 ) such that a_1 +a_2 sin x+a_3 cos x+a_4 sin 2x+a_5 cos 2x=0 holds for all x is

$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{5}−\mathrm{tuples} \\ $$$$\left({a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} ,\:{a}_{\mathrm{4}} ,\:{a}_{\mathrm{5}} \right)\:\mathrm{such}\:\mathrm{that}\: \\ $$$${a}_{\mathrm{1}} +{a}_{\mathrm{2}} \mathrm{sin}\:{x}+{a}_{\mathrm{3}} \mathrm{cos}\:{x}+{a}_{\mathrm{4}} \mathrm{sin}\:\mathrm{2}{x}+{a}_{\mathrm{5}} \mathrm{cos}\:\mathrm{2}{x}=\mathrm{0} \\ $$$$\mathrm{holds}\:\mathrm{for}\:\mathrm{all}\:\:\:{x}\:\:\mathrm{is} \\ $$

Question Number 44120 Answers: 1 Comments: 0

Question Number 44117 Answers: 0 Comments: 0

Question Number 44103 Answers: 1 Comments: 1

Question Number 44098 Answers: 3 Comments: 0

Question Number 44097 Answers: 3 Comments: 0

Question Number 44096 Answers: 2 Comments: 0

Question Number 44095 Answers: 1 Comments: 0

Question Number 44094 Answers: 2 Comments: 2

Question Number 44092 Answers: 0 Comments: 1

Question Number 44091 Answers: 1 Comments: 0

Question Number 44089 Answers: 0 Comments: 0

Question Number 44088 Answers: 1 Comments: 0

Question Number 44085 Answers: 0 Comments: 6

lim x→0 [((sin ∣x∣)/x)]

$${lim}\:\mathrm{x}\rightarrow\mathrm{0}\:\left[\frac{\mathrm{sin}\:\mid{x}\mid}{{x}}\right] \\ $$

Question Number 44069 Answers: 1 Comments: 1

∫(dx/((x+1)(√(x^2 +2)))) = ?

$$\int\frac{{dx}}{\left({x}+\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}}\:=\:? \\ $$

Question Number 44068 Answers: 2 Comments: 0

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