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Question Number 26360    Answers: 0   Comments: 1

find the value of ∫_0 ^( ∝ ) ((cos(αx))/(1+x^2 )) dx .

$$\:\:{find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{0}} ^{\:\propto\:} \:\frac{{cos}\left(\alpha{x}\right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:\:. \\ $$

Question Number 26352    Answers: 1   Comments: 0

x(x+9)=(x+3)(x+7)−10

$${x}\left({x}+\mathrm{9}\right)=\left({x}+\mathrm{3}\right)\left({x}+\mathrm{7}\right)−\mathrm{10} \\ $$

Question Number 26348    Answers: 1   Comments: 1

Question Number 26347    Answers: 0   Comments: 0

solve: 5x(1 + (1/(x^2 + y^2 ))) = 12 ..... equation (i) 5y(1 − (1/(x^2 + y^2 ))) = 4 ...... equation (ii)

$$\mathrm{solve}: \\ $$$$\mathrm{5x}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\right)\:=\:\mathrm{12}\:\:\:\:\:\:\:\:.....\:\mathrm{equation}\:\left(\mathrm{i}\right) \\ $$$$\mathrm{5y}\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\right)\:=\:\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:......\:\mathrm{equation}\:\left(\mathrm{ii}\right) \\ $$

Question Number 26359    Answers: 0   Comments: 2

find lim_(n−>∝) ∫_0 ^n (1−(t/n))^(n−1) dt .

$${find}\:\:\:{lim}_{{n}−>\propto} \:\:\int_{\mathrm{0}} ^{{n}} \:\left(\mathrm{1}−\frac{{t}}{{n}}\right)^{{n}−\mathrm{1}} {dt}\:\:\:. \\ $$

Question Number 26358    Answers: 1   Comments: 0

find the value of ∫_0 ^(π/4) tan^n x dx with n element from N.

$${find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:{tan}^{{n}} {x}\:{dx}\:\:\:\:{with}\: \\ $$$${n}\:\:{element}\:{from}\:\mathbb{N}. \\ $$

Question Number 26357    Answers: 1   Comments: 1

find the value of ∫_0 ^(π/2) ln(cosθ)dθ and ∫_0 ^(π/2) ln(sinθ)dθ .

$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left({cos}\theta\right){d}\theta\:\:{and}\: \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{ln}\left({sin}\theta\right){d}\theta\:\:\:. \\ $$

Question Number 26329    Answers: 1   Comments: 0

A small particle moving with a uniform acceleration a covers distances X and Y in the first two equal and consecutive intervals of time t. Show that a = ((Y − X)/t^2 )

$$\mathrm{A}\:\mathrm{small}\:\mathrm{particle}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{a}\:\mathrm{uniform}\:\mathrm{acceleration}\:\mathrm{a}\:\mathrm{covers}\:\mathrm{distances}\: \\ $$$$\mathrm{X}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{in}\:\mathrm{the}\:\mathrm{first}\:\mathrm{two}\:\mathrm{equal}\:\mathrm{and}\:\mathrm{consecutive}\:\mathrm{intervals}\:\mathrm{of}\:\mathrm{time}\:\mathrm{t}.\:\mathrm{Show}\:\mathrm{that} \\ $$$$\mathrm{a}\:=\:\frac{\mathrm{Y}\:−\:\mathrm{X}}{\mathrm{t}^{\mathrm{2}} } \\ $$

Question Number 26328    Answers: 1   Comments: 1

Three towns X, Y and Z are on a straight road and Y is the mid−way between X and Z. A motor cyclist moving with uniform acceleration passes X, Y and Z. The speed with which the motocyclist passes X and Z are 20m/s and 40m/s respectively. Find the speed with which the motorcyclist passes Y.

$$\mathrm{Three}\:\mathrm{towns}\:\mathrm{X},\:\mathrm{Y}\:\mathrm{and}\:\mathrm{Z}\:\mathrm{are}\:\mathrm{on}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{road}\:\mathrm{and}\:\mathrm{Y}\:\mathrm{is}\:\mathrm{the}\:\mathrm{mid}−\mathrm{way}\:\mathrm{between} \\ $$$$\mathrm{X}\:\mathrm{and}\:\mathrm{Z}.\:\mathrm{A}\:\mathrm{motor}\:\mathrm{cyclist}\:\mathrm{moving}\:\mathrm{with}\:\mathrm{uniform}\:\mathrm{acceleration}\:\mathrm{passes}\:\mathrm{X},\:\mathrm{Y}\:\mathrm{and}\:\mathrm{Z}.\: \\ $$$$\mathrm{The}\:\mathrm{speed}\:\mathrm{with}\:\mathrm{which}\:\mathrm{the}\:\mathrm{motocyclist}\:\mathrm{passes}\:\mathrm{X}\:\mathrm{and}\:\mathrm{Z}\:\mathrm{are}\:\mathrm{20m}/\mathrm{s}\:\mathrm{and}\:\mathrm{40m}/\mathrm{s}\: \\ $$$$\mathrm{respectively}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{speed}\:\mathrm{with}\:\mathrm{which}\:\mathrm{the}\:\mathrm{motorcyclist}\:\mathrm{passes}\:\mathrm{Y}. \\ $$

Question Number 26324    Answers: 1   Comments: 3

Question Number 26320    Answers: 2   Comments: 3

solve x^2 −1=2^x

$$\mathrm{solve}\:{x}^{\mathrm{2}} −\mathrm{1}=\mathrm{2}^{{x}} \\ $$

Question Number 26316    Answers: 1   Comments: 0

A rectangular hall 12m long and 10m broad, is surrounded by a path 2m wide. Find the area of the path.

$$\mathrm{A}\:\mathrm{rectangular}\:\mathrm{hall}\:\mathrm{12m}\:\mathrm{long}\:\mathrm{and}\:\mathrm{10m} \\ $$$$\mathrm{broad},\:\mathrm{is}\:\mathrm{surrounded}\:\mathrm{by}\:\mathrm{a}\:\mathrm{path}\:\mathrm{2m} \\ $$$$\mathrm{wide}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{the}\:\mathrm{path}. \\ $$

Question Number 26312    Answers: 0   Comments: 6

Question Number 26300    Answers: 0   Comments: 1

y=a^(arctg(√x)) y′=?

$${y}={a}^{\mathrm{arc}{tg}\sqrt{{x}}} \\ $$$${y}'=? \\ $$

Question Number 26299    Answers: 0   Comments: 1

y=((1+e^x )/(1−e^x )) y′=?

$${y}=\frac{\mathrm{1}+{e}^{{x}} }{\mathrm{1}−{e}^{{x}} } \\ $$$${y}'=? \\ $$

Question Number 26298    Answers: 0   Comments: 2

y=sin^4 x y′=? differential

$${y}=\mathrm{sin}\:^{\mathrm{4}} {x} \\ $$$${y}'=?\:{differential} \\ $$

Question Number 26297    Answers: 1   Comments: 0

y=x−(2/x^4 )−(1/(3x^3 )) y′=?

$${y}={x}−\frac{\mathrm{2}}{{x}^{\mathrm{4}} }−\frac{\mathrm{1}}{\mathrm{3}{x}^{\mathrm{3}} } \\ $$$${y}'=? \\ $$

Question Number 26294    Answers: 0   Comments: 1

lim_(x→0) (((√(1+xsin x ))−(√(cos 2x)))/(tg^2 (x/2)))

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{1}+{x}\mathrm{sin}\:{x}\:}−\sqrt{\mathrm{cos}\:\mathrm{2}{x}}}{{tg}^{\mathrm{2}} \frac{{x}}{\mathrm{2}}} \\ $$

Question Number 26293    Answers: 1   Comments: 0

lim_(x→∞) (3(√(1−x^3 +x)))

$$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{3}\sqrt{\left.\mathrm{1}−{x}^{\mathrm{3}} +{x}\right)}\right. \\ $$

Question Number 26290    Answers: 0   Comments: 1

f(x)=4x^2 +1 x_0 =2

$${f}\left({x}\right)=\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}\: \\ $$$${x}_{\mathrm{0}} =\mathrm{2} \\ $$

Question Number 26289    Answers: 1   Comments: 0

lim_(x→2) ((1/(x−2))−((12)/(x^3 −8)))

$$\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\left(\frac{\mathrm{1}}{{x}−\mathrm{2}}−\frac{\mathrm{12}}{{x}^{\mathrm{3}} −\mathrm{8}}\right) \\ $$

Question Number 26288    Answers: 2   Comments: 0

(d/dx) ((x−4)/(2(√x)))

$$\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\:\frac{\boldsymbol{{x}}−\mathrm{4}}{\mathrm{2}\sqrt{\boldsymbol{{x}}}} \\ $$

Question Number 26313    Answers: 0   Comments: 0

If domain of y = f(x) is [−3, 2] and g(x) = f(∣[x]∣) ([∙] denotes the greatest integer function), then domain of g(x) is

$$\mathrm{If}\:\mathrm{domain}\:\mathrm{of}\:{y}\:=\:{f}\left({x}\right)\:\mathrm{is}\:\left[−\mathrm{3},\:\mathrm{2}\right]\:\mathrm{and} \\ $$$${g}\left({x}\right)\:=\:{f}\left(\mid\left[{x}\right]\mid\right)\:\left(\left[\centerdot\right]\:\mathrm{denotes}\:\mathrm{the}\:\mathrm{greatest}\right. \\ $$$$\left.\mathrm{integer}\:\mathrm{function}\right),\:\mathrm{then}\:\mathrm{domain}\:\mathrm{of}\:{g}\left({x}\right) \\ $$$$\mathrm{is} \\ $$

Question Number 26354    Answers: 1   Comments: 0

Question Number 26281    Answers: 1   Comments: 0

If f(x) is a function satisfying f(x+y)=f(x) f(y) for all x, y ∈ N such that f(1)=3 and Σ_(x=1) ^n f(x)=120. Then the value of n is

$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathrm{satisfying} \\ $$$$\:{f}\left({x}+{y}\right)={f}\left({x}\right)\:{f}\left({y}\right)\:\mathrm{for}\:\mathrm{all}\:{x},\:{y}\:\in\:{N}\:\:\mathrm{such} \\ $$$$\mathrm{that}\:{f}\left(\mathrm{1}\right)=\mathrm{3}\:\mathrm{and}\:\underset{{x}=\mathrm{1}} {\overset{{n}} {\sum}}\:{f}\left({x}\right)=\mathrm{120}.\:\mathrm{Then} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}\:\mathrm{is} \\ $$

Question Number 26274    Answers: 0   Comments: 2

could there be an analytical or numerical meghod for solving this non-linear simultaneous equation x+y=5 x^x +y^y =31 please help if possible

$${could}\:{there}\:{be}\:{an}\:{analytical}\:{or} \\ $$$${numerical}\:{meghod}\:{for}\:{solving} \\ $$$${this}\:{non}-{linear}\:{simultaneous} \\ $$$${equation} \\ $$$${x}+{y}=\mathrm{5} \\ $$$${x}^{{x}} +{y}^{{y}} =\mathrm{31} \\ $$$$ \\ $$$${please}\:{help}\:{if}\:{possible} \\ $$

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