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Question Number 75166 Answers: 1 Comments: 0

Question Number 75158 Answers: 1 Comments: 0

Question Number 75156 Answers: 1 Comments: 0

Question Number 75147 Answers: 1 Comments: 0

hello show that sin((5π)/(18))cos((13π)/(18))=−(1/2)sin((4π)/9)

$$\mathrm{hello} \\ $$$$\mathrm{show}\:\mathrm{that}\: \\ $$$$\mathrm{sin}\frac{\mathrm{5}\pi}{\mathrm{18}}\mathrm{cos}\frac{\mathrm{13}\pi}{\mathrm{18}}=−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\frac{\mathrm{4}\pi}{\mathrm{9}} \\ $$

Question Number 75141 Answers: 1 Comments: 2

Question Number 75136 Answers: 1 Comments: 3

Question Number 75131 Answers: 1 Comments: 0

please help me to show that tan^2 ((π/8))+2tan((π/8))−1=0

$$\mathrm{please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{to}\:\mathrm{show}\:\mathrm{that} \\ $$$$\mathrm{tan}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{8}}\right)+\mathrm{2tan}\left(\frac{\pi}{\mathrm{8}}\right)−\mathrm{1}=\mathrm{0} \\ $$

Question Number 75124 Answers: 0 Comments: 0

Question Number 75122 Answers: 0 Comments: 4

Question Number 75110 Answers: 1 Comments: 0

Question Number 75105 Answers: 0 Comments: 1

Question Number 75104 Answers: 1 Comments: 0

Question Number 75102 Answers: 0 Comments: 0

show by integration that the centroid of a semi−circular lamina of radius a from the centre is ((4a)/(3π)).

$$\mathrm{show}\:\mathrm{by}\:\mathrm{integration}\:\mathrm{that}\:\mathrm{the}\:\mathrm{centroid} \\ $$$$\mathrm{of}\:\mathrm{a}\:\mathrm{semi}−\mathrm{circular}\:\mathrm{lamina}\:\mathrm{of}\:\mathrm{radius}\:{a}\: \\ $$$$\mathrm{from}\:\mathrm{the}\:\mathrm{centre}\:\mathrm{is}\:\:\:\frac{\mathrm{4}{a}}{\mathrm{3}\pi}. \\ $$

Question Number 75101 Answers: 4 Comments: 2

the vector equations of two lines L_1 and L_2 is given by L_1 :r= i−j+3k + λ(i−j +k) L_2 : r= 2i+aj + 6k + μ(2i + j + 3k) where a,λ,μ are real constants. given that L_1 and L_2 intersect find a. the value of the constant a. b. the position vector of the point of intersection between L_1 and L_2 c. the cosine of the acute angle between L_1 and L_2 please help

$${the}\:{vector}\:{equations}\:{of}\:{two}\:{lines}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \:{is}\:{given}\:{by} \\ $$$$\:{L}_{\mathrm{1}} :{r}=\:\boldsymbol{{i}}−\boldsymbol{{j}}+\mathrm{3}\boldsymbol{{k}}\:+\:\lambda\left(\boldsymbol{{i}}−\boldsymbol{{j}}\:+\boldsymbol{{k}}\right) \\ $$$${L}_{\mathrm{2}} \::\:{r}=\:\mathrm{2}\boldsymbol{{i}}+{a}\boldsymbol{{j}}\:+\:\mathrm{6}\boldsymbol{{k}}\:+\:\mu\left(\mathrm{2}\boldsymbol{{i}}\:+\:\boldsymbol{{j}}\:+\:\mathrm{3}\boldsymbol{{k}}\right) \\ $$$${where}\:{a},\lambda,\mu\:{are}\:{real}\:{constants}. \\ $$$${given}\:{that}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \:{intersect}\:{find} \\ $$$${a}.\:\:{the}\:{value}\:{of}\:{the}\:{constant}\:{a}. \\ $$$${b}.\:\:{the}\:{position}\:{vector}\:{of}\:{the}\:{point}\:{of}\: \\ $$$${intersection}\:{between}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \\ $$$${c}.\:{the}\:{cosine}\:{of}\:{the}\:{acute}\:{angle}\:{between}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \\ $$$${please}\:{help} \\ $$$$ \\ $$

Question Number 75100 Answers: 2 Comments: 0

find the intervals cor which the function h(x) = x^3 −3x is a) strickly increasing b) strickly decreasing

$$\:{find}\:{the}\:{intervals}\:{cor}\:{which}\:{the}\:{function} \\ $$$${h}\left({x}\right)\:=\:{x}^{\mathrm{3}} −\mathrm{3}{x}\:{is} \\ $$$$\left.{a}\right)\:{strickly}\:{increasing} \\ $$$$\left.{b}\right)\:{strickly}\:{decreasing} \\ $$$$ \\ $$

Question Number 75099 Answers: 1 Comments: 4

Given the matrix A = ((1,(−1),1),(0,2,( −1)),(2,3,0) ) and B= ((3,3,(−1)),((−2),(−2),1),((−4),(−5),2) ) find the matrix product AB and BA state the relationship between A and B find also the matrix product BM, where M= ((8),((−7)),(1) ) Hence solve the system of equations: x−y + z = 8, 2y −z =−7, 2x + 3y = 1.

$${Given}\:{the}\:{matrix}\: \\ $$$${A}\:=\:\begin{pmatrix}{\mathrm{1}}&{−\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{2}}&{\:−\mathrm{1}}\\{\mathrm{2}}&{\mathrm{3}}&{\mathrm{0}}\end{pmatrix}\:\:{and}\:{B}=\:\begin{pmatrix}{\mathrm{3}}&{\mathrm{3}}&{−\mathrm{1}}\\{−\mathrm{2}}&{−\mathrm{2}}&{\mathrm{1}}\\{−\mathrm{4}}&{−\mathrm{5}}&{\mathrm{2}}\end{pmatrix} \\ $$$${find}\:{the}\:{matrix}\:{product}\:{AB}\:{and}\:{BA} \\ $$$${state}\:{the}\:{relationship}\:{between}\:{A}\:{and}\:{B} \\ $$$${find}\:{also}\:{the}\:{matrix}\:{product}\:{BM},\:{where}\:{M}=\begin{pmatrix}{\mathrm{8}}\\{−\mathrm{7}}\\{\mathrm{1}}\end{pmatrix} \\ $$$${Hence}\:{solve}\:{the}\:{system}\:{of}\:{equations}: \\ $$$$\:\:{x}−{y}\:+\:{z}\:=\:\mathrm{8}, \\ $$$$\:\:\:\:\:\:\:\mathrm{2}{y}\:−{z}\:=−\mathrm{7}, \\ $$$$\:\:\mathrm{2}{x}\:+\:\mathrm{3}{y}\:=\:\mathrm{1}. \\ $$

Question Number 75098 Answers: 1 Comments: 0

the function f is defined by f(x) = (2/(x^2 −1)) a) Express f into partial fraction b.show that ∫_3 ^5 f(x) dx = ln((4/3))

$$\mathrm{the}\:\mathrm{function}\:\mathrm{f}\:\mathrm{is}\:\mathrm{defined}\:\mathrm{by}\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}}{{x}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.{a}\right)\:\mathrm{Express}\:\mathrm{f}\:\mathrm{into}\:\mathrm{partial}\:\mathrm{fraction} \\ $$$$\mathrm{b}.\mathrm{show}\:\mathrm{that}\:\int_{\mathrm{3}} ^{\mathrm{5}} {f}\left({x}\right)\:{dx}\:=\:{ln}\left(\frac{\mathrm{4}}{\mathrm{3}}\right) \\ $$

Question Number 75093 Answers: 1 Comments: 1

∫cos^3 xsin^3 xdx

$$\int{cos}^{\mathrm{3}} {xsin}^{\mathrm{3}} {xdx} \\ $$

Question Number 75083 Answers: 1 Comments: 0

A function f is given by f(x) = { ((x^2 −3, 0≤x<2)),((4x−7, 2≤x<4)) :} is such that f(x) = f(x + 4) find f(27) and f(−106).

$${A}\:{function}\:{f}\:{is}\:{given}\:{by}\: \\ $$$$\:{f}\left({x}\right)\:=\:\begin{cases}{{x}^{\mathrm{2}} −\mathrm{3},\:\:\mathrm{0}\leqslant{x}<\mathrm{2}}\\{\mathrm{4}{x}−\mathrm{7},\:\mathrm{2}\leqslant{x}<\mathrm{4}}\end{cases} \\ $$$${is}\:{such}\:{that}\:{f}\left({x}\right)\:=\:{f}\left({x}\:+\:\mathrm{4}\right)\: \\ $$$${find}\:\:{f}\left(\mathrm{27}\right)\:{and}\:{f}\left(−\mathrm{106}\right). \\ $$

Question Number 75082 Answers: 1 Comments: 0

find ∫_0 ^π e^(cosx) sinx dx

$${find} \\ $$$$\int_{\mathrm{0}} ^{\pi} {e}^{{cosx}} {sinx}\:{dx} \\ $$

Question Number 75081 Answers: 1 Comments: 0

Evaluate ∫_1 ^(3 ) (x^2 /(1+x)) dx

$${Evaluate}\: \\ $$$$\:\int_{\mathrm{1}} ^{\mathrm{3}\:} \frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}}\:{dx} \\ $$

Question Number 75080 Answers: 0 Comments: 1

Find out A=Σ_(n=0) ^∞ ∫_0 ^(π/2) (1−(√(sinx)))^n cosxdx

$$\mathrm{Find}\:\mathrm{out}\: \\ $$$$\mathrm{A}=\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\sqrt{\mathrm{sinx}}\right)^{\mathrm{n}} \mathrm{cosxdx} \\ $$

Question Number 75079 Answers: 0 Comments: 0

Let f∈C([0,1],[0,1]) Prove that lim_(n→∞) ∫_([0,1]^n ) f((1/n)Σ_(i=1) ^n x_i )dx_1 ....dx_n =f((1/2))

$$\mathrm{Let}\:\mathrm{f}\in\mathrm{C}\left(\left[\mathrm{0},\mathrm{1}\right],\left[\mathrm{0},\mathrm{1}\right]\right)\:\: \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\:\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{n}} } \mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{n}}\underset{\mathrm{i}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{x}_{\mathrm{i}} \:\right)\mathrm{dx}_{\mathrm{1}} ....\mathrm{dx}_{\mathrm{n}} \:=\mathrm{f}\left(\frac{\mathrm{1}}{\mathrm{2}}\right) \\ $$

Question Number 75078 Answers: 1 Comments: 0

Question Number 75066 Answers: 1 Comments: 0

Prove that if f is a function R→R and there exist x_0 >0 , such as L(f)(x_0 ) exist then lim_(t→∞) f(t)e^(−x_0 t) =0 and ∀ x>x_0 L(f)(x) exist. L(f) is the Laplace transformed function

$$\mathrm{Prove}\:\:\mathrm{that}\:\mathrm{if}\:\:\mathrm{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{function}\:\mathbb{R}\rightarrow\mathbb{R}\: \\ $$$$\mathrm{and}\:\:\mathrm{there}\:\mathrm{exist}\:\mathrm{x}_{\mathrm{0}} >\mathrm{0}\:\:,\:\mathrm{such}\:\mathrm{as}\:\:\mathrm{L}\left(\mathrm{f}\right)\left(\mathrm{x}_{\mathrm{0}} \right)\:\mathrm{exist}\: \\ $$$$\mathrm{then}\:\underset{\mathrm{t}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{f}\left(\mathrm{t}\right)\mathrm{e}^{−\mathrm{x}_{\mathrm{0}} \mathrm{t}} =\mathrm{0}\:\mathrm{and}\:\forall\:\mathrm{x}>\mathrm{x}_{\mathrm{0}} \:\:\mathrm{L}\left(\mathrm{f}\right)\left(\mathrm{x}\right)\:\mathrm{exist}. \\ $$$$\mathrm{L}\left(\mathrm{f}\right)\:\mathrm{is}\:\mathrm{the}\:\mathrm{Laplace}\:\mathrm{transformed}\:\mathrm{function} \\ $$

Question Number 75063 Answers: 1 Comments: 1

calculate Σ_(n=1) ^(17) (((−1)^n )/n^3 )

$${calculate}\:\:\sum_{{n}=\mathrm{1}} ^{\mathrm{17}} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} } \\ $$

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