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Question Number 42709 Answers: 1 Comments: 3

If f(x)= x^3 −((3x^2 )/2) +x + (1/4). Then ∫_(1/4) ^(3/4) f(f(x))dx =?

$$\mathrm{If}\:\mathrm{f}\left({x}\right)=\:{x}^{\mathrm{3}} \:−\frac{\mathrm{3}{x}^{\mathrm{2}} }{\mathrm{2}}\:+{x}\:+\:\frac{\mathrm{1}}{\mathrm{4}}. \\ $$$${T}\mathrm{hen}\:\int_{\frac{\mathrm{1}}{\mathrm{4}}} ^{\frac{\mathrm{3}}{\mathrm{4}}} \:\mathrm{f}\left(\mathrm{f}\left({x}\right)\right)\mathrm{d}{x}\:=? \\ $$

Question Number 42704 Answers: 0 Comments: 4

f(x) = ((e^(3x) +e^(−3x) )/2) 1) determine f^(−1) (x) 2) calculate ∫_0 ^1 x f(x)dx and ∫_0 ^1 f(x)dx 3) calculate ∫ f^(−1) (x)dx 4) calculate u_n = ∫_0 ^π f(x)cos(nx)dx and v_n = ∫_0 ^n f(x)sin(nx)dx find nature of Σ (v_n /u_n ) ∫_0 ^1 xf(x) dx =(1/2) ∫_0 ^1 x e^(3x) dx +(1/2) ∫_0 ^1 x e^(−3x) dx (by parts) =(1/2){ [(x/3)e^(3x) ]_0 ^1 −(1/3)∫_0 ^1 e^(3x) dx +[−(x/3)e^(−3x) ]_0 ^1 +(1/3)∫_0 ^1 e^(−3x) dx} =(1/2){(e^3 /3) −(1/9)(e^3 −1) −(e^(−3) /3) −(1/9)(e^(−3) −1)}

$${f}\left({x}\right)\:\:=\:\:\frac{{e}^{\mathrm{3}{x}} \:+{e}^{−\mathrm{3}{x}} }{\mathrm{2}} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:{x}\:{f}\left({x}\right){dx}\:\:\:\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:\:{calculate}\:\:\:\int\:\:{f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{calculate}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:{f}\left({x}\right){cos}\left({nx}\right){dx}\:{and}\:{v}_{{n}} =\:\int_{\mathrm{0}} ^{{n}} \:\:{f}\left({x}\right){sin}\left({nx}\right){dx} \\ $$$${find}\:{nature}\:{of}\:\Sigma\:\frac{{v}_{{n}} }{{u}_{{n}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{xf}\left({x}\right)\:{dx}\:=\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{\mathrm{3}{x}} {dx}\:+\frac{\mathrm{1}}{\mathrm{2}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}\:{e}^{−\mathrm{3}{x}} {dx}\:\:\:\left({by}\:{parts}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\left[\frac{{x}}{\mathrm{3}}{e}^{\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:−\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{\mathrm{3}{x}} {dx}\:\:+\left[−\frac{{x}}{\mathrm{3}}{e}^{−\mathrm{3}{x}} \right]_{\mathrm{0}} ^{\mathrm{1}} \:+\frac{\mathrm{1}}{\mathrm{3}}\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{e}^{−\mathrm{3}{x}} {dx}\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\frac{{e}^{\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{\mathrm{3}} −\mathrm{1}\right)\:−\frac{{e}^{−\mathrm{3}} }{\mathrm{3}}\:−\frac{\mathrm{1}}{\mathrm{9}}\left({e}^{−\mathrm{3}} −\mathrm{1}\right)\right\} \\ $$$$ \\ $$

Question Number 42695 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) (x^4 /(x^(8 ) +16))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\:\:\:\frac{{x}^{\mathrm{4}} }{{x}^{\mathrm{8}\:} \:+\mathrm{16}}{dx} \\ $$

Question Number 42689 Answers: 0 Comments: 3

let f(x) = (x/(x^3 −2x +1)) 1) find D_f 2) find f^((n)) (x) then f^((n)) (0) 3) developp f at integr serie.

$${let}\:{f}\left({x}\right)\:=\:\frac{{x}}{{x}^{\mathrm{3}} −\mathrm{2}{x}\:\:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{f}^{\left({n}\right)} \left({x}\right)\:\:{then}\:\:{f}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{f}\:{at}\:{integr}\:{serie}. \\ $$

Question Number 42688 Answers: 0 Comments: 3

let g(x) =((x−1)/(x^2 +x +1)) 1) find g^((n)) (x) 2)calculate g^((n)) (0) 3) developp g at integr serie.

$${let}\:{g}\left({x}\right)\:=\frac{{x}−\mathrm{1}}{{x}^{\mathrm{2}} +{x}\:+\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:\:{find}\:{g}^{\left({n}\right)} \left({x}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:{g}^{\left({n}\right)} \left(\mathrm{0}\right) \\ $$$$\left.\mathrm{3}\right)\:{developp}\:{g}\:{at}\:\:{integr}\:{serie}. \\ $$

Question Number 42684 Answers: 0 Comments: 2

The coefficient of x^4 in the expansion of ((x/2) − (3/x^2 ))^(10) is

$$\mathrm{The}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{\mathrm{4}} \:\mathrm{in}\:\mathrm{the}\:\mathrm{expansion}\:\mathrm{of} \\ $$$$\left(\frac{{x}}{\mathrm{2}}\:−\:\frac{\mathrm{3}}{{x}^{\mathrm{2}} }\right)^{\mathrm{10}} \:\mathrm{is} \\ $$

Question Number 42681 Answers: 0 Comments: 0

Question Number 42680 Answers: 0 Comments: 2

calculale A_n (α) = ∫_(−∞) ^(+∞) ((cos(αx^n ))/(1+x^2 )) dx with n integr natural.

$${calculale}\:\:{A}_{{n}} \left(\alpha\right)\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{cos}\left(\alpha{x}^{{n}} \right)}{\mathrm{1}+{x}^{\mathrm{2}} }\:{dx}\:{with} \\ $$$${n}\:{integr}\:{natural}. \\ $$$$ \\ $$

Question Number 42679 Answers: 0 Comments: 2

calculate ∫_(π/4) ^(π/3) ((sinx)/(cosx +tanx))dx .

$${calculate}\:\:\:\:\int_{\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{3}}} \:\:\:\:\:\:\frac{{sinx}}{{cosx}\:+{tanx}}{dx}\:. \\ $$

Question Number 42673 Answers: 1 Comments: 0

Question Number 42672 Answers: 0 Comments: 0

Simplify: (x + y + z)(x^(−1) + y^(−1) + z^(−1) ) = (x^(−1) y^(−1) z^(−1) )(x + y)(y + z)(z + x)

$$\mathrm{Simplify}:\:\:\:\left(\mathrm{x}\:+\:\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{x}^{−\mathrm{1}} \:+\:\mathrm{y}^{−\mathrm{1}} \:+\:\mathrm{z}^{−\mathrm{1}} \right)\:=\:\left(\mathrm{x}^{−\mathrm{1}} \:\mathrm{y}^{−\mathrm{1}} \:\mathrm{z}^{−\mathrm{1}} \right)\left(\mathrm{x}\:+\:\mathrm{y}\right)\left(\mathrm{y}\:+\:\mathrm{z}\right)\left(\mathrm{z}\:+\:\mathrm{x}\right) \\ $$

Question Number 42671 Answers: 0 Comments: 0

If pqr = 1 Hence evaluate: (1/(1 + e + f^(−1) )) + (1/(1 + f + g^(−1) )) + (1/(1 + g + e^(−1) ))

$$\mathrm{If}\:\mathrm{pqr}\:=\:\mathrm{1} \\ $$$$\mathrm{Hence}\:\mathrm{evaluate}:\:\:\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{e}\:+\:\mathrm{f}^{−\mathrm{1}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{f}\:+\:\mathrm{g}^{−\mathrm{1}} }\:\:+\:\:\frac{\mathrm{1}}{\mathrm{1}\:+\:\mathrm{g}\:+\:\mathrm{e}^{−\mathrm{1}} } \\ $$

Question Number 42670 Answers: 1 Comments: 3

If a, b and c are in a GP. Prove that log_n a , log_n b , log_n c are in AP

$$\mathrm{If}\:\:\mathrm{a},\:\mathrm{b}\:\mathrm{and}\:\mathrm{c}\:\:\mathrm{are}\:\mathrm{in}\:\mathrm{a}\:\mathrm{GP}.\:\:\mathrm{Prove}\:\mathrm{that}\:\:\:\mathrm{log}_{\mathrm{n}} \mathrm{a}\:,\:\:\mathrm{log}_{\mathrm{n}} \mathrm{b}\:\:,\:\:\mathrm{log}_{\mathrm{n}} \mathrm{c}\:\:\:\mathrm{are}\:\mathrm{in}\:\mathrm{AP} \\ $$

Question Number 42698 Answers: 1 Comments: 0

A boy lying flat on level ground sees a bird on a tree and the angle of Elevation from the boy to the birth is 42°,if the boy is 6m from the tree.find the hieght of the tree if the bird is at the top of the tree

$${A}\:{boy}\:{lying}\:{flat}\:{on}\:{level}\:{ground}\:{sees}\:{a}\:{bird}\:\:{on}\:{a}\:{tree}\:{and}\:{the} \\ $$$${angle}\:{of}\:{Elevation}\:{from}\:{the}\:{boy}\:{to}\:{the}\:{birth}\:{is}\:\mathrm{42}°,{if}\:{the}\:{boy} \\ $$$${is}\:\mathrm{6}{m}\:{from}\:{the}\:{tree}.{find}\:{the}\:{hieght}\:{of}\:{the}\:{tree}\:{if}\:{the}\:{bird} \\ $$$${is}\:{at}\:{the}\:{top}\:{of}\:{the}\:{tree} \\ $$

Question Number 42668 Answers: 0 Comments: 0

If ∣a sin^2 θ+b sin θ cos θ+c cos^2 θ−(1/2)(a−c)∣ ≤ (1/2) k, then k^2 is equal to

$$\mathrm{If} \\ $$$$\mid{a}\:\mathrm{sin}^{\mathrm{2}} \theta+{b}\:\mathrm{sin}\:\theta\:\mathrm{cos}\:\theta+{c}\:\mathrm{cos}^{\mathrm{2}} \theta−\frac{\mathrm{1}}{\mathrm{2}}\left({a}−{c}\right)\mid \\ $$$$\:\:\:\leqslant\:\frac{\mathrm{1}}{\mathrm{2}}\:{k},\:\mathrm{then}\:{k}^{\mathrm{2}} \:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$

Question Number 42708 Answers: 0 Comments: 1

calculate f(α)=∫_0 ^∞ ((e^(−2x) −e^(−x) )/x^2 ) e^(−αx^2 ) dx with α>0 1) find the value of ∫_0 ^∞ ((e^(−2x) −e^(−x) )/x^2 ) e^(−2x^2 ) dx

$${calculate}\:\:{f}\left(\alpha\right)=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\mathrm{2}{x}} −{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\alpha{x}^{\mathrm{2}} } {dx}\:\:{with}\:\alpha>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{the}\:{value}\:{of}\:\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{e}^{−\mathrm{2}{x}} \:−{e}^{−{x}} }{{x}^{\mathrm{2}} }\:{e}^{−\mathrm{2}{x}^{\mathrm{2}} } {dx} \\ $$

Question Number 42663 Answers: 0 Comments: 0

if 1.225 g of KClO_3 are heated. Calculate the mass of potassium Chloride produced. Detemine the volume of oxygen obtained at r.t.p

$${if}\:\mathrm{1}.\mathrm{225}\:{g}\:{of}\:{KClO}_{\mathrm{3}} \:{are}\:{heated}. \\ $$$${Calculate}\:{the}\:{mass}\:{of}\:{potassium}\:{Chloride}\:{produced}. \\ $$$${Detemine}\:{the}\:{volume}\:{of}\:{oxygen}\:{obtained}\:{at}\:{r}.{t}.{p} \\ $$

Question Number 42657 Answers: 0 Comments: 0

Question Number 42656 Answers: 1 Comments: 0

Question Number 42654 Answers: 2 Comments: 2

Question Number 42650 Answers: 0 Comments: 0

Question Number 42649 Answers: 0 Comments: 0

Question Number 42648 Answers: 0 Comments: 0

Question Number 42639 Answers: 0 Comments: 0

Find LCM [((13)/2),(2/(13)),(4/7)] [((57)),(0) ]×

$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Find}\:\mathrm{LCM}\:\left[\frac{\mathrm{13}}{\mathrm{2}},\frac{\mathrm{2}}{\mathrm{13}},\frac{\mathrm{4}}{\mathrm{7}}\right] \\ $$$$\begin{bmatrix}{\mathrm{57}}\\{\mathrm{0}}\end{bmatrix}× \\ $$

Question Number 42636 Answers: 0 Comments: 1

lim_(n→∞) Σ_(r=1) ^n (r/(n^2 +n+r))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{r}}{{n}^{\mathrm{2}} +{n}+{r}} \\ $$$$ \\ $$

Question Number 42631 Answers: 1 Comments: 7

let f(x)=2(√(x−1)) −2x 1) find D_f 2) study the variation of f(x) 3 ) calculate ∫_1 ^3 f(x)dx 4) determine f^(−1) (x) and calculate ∫_1 ^3 f^(−1) (x)dx 5) find the values of A = ∫_1 ^3 ((f(x))/(f^(−1) (x)dx)) and B = ((∫_1 ^3 f(x))/(∫_1 ^3 f^(−1) (x))) dx.

$${let}\:{f}\left({x}\right)=\mathrm{2}\sqrt{{x}−\mathrm{1}}\:−\mathrm{2}{x} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{D}_{{f}} \\ $$$$\left.\mathrm{2}\right)\:{study}\:{the}\:{variation}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{3}\:\right)\:{calculate}\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}\left({x}\right){dx} \\ $$$$\left.\mathrm{4}\right)\:{determine}\:{f}^{−\mathrm{1}} \left({x}\right)\:{and}\:{calculate}\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}^{−\mathrm{1}} \left({x}\right){dx} \\ $$$$\left.\mathrm{5}\right)\:\:{find}\:{the}\:{values}\:{of}\:\:{A}\:=\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:\:\:\frac{{f}\left({x}\right)}{{f}^{−\mathrm{1}} \left({x}\right){dx}}\:{and}\: \\ $$$${B}\:=\:\frac{\int_{\mathrm{1}} ^{\mathrm{3}} \:\:{f}\left({x}\right)}{\int_{\mathrm{1}} ^{\mathrm{3}} \:{f}^{−\mathrm{1}} \left({x}\right)}\:{dx}. \\ $$

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