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IntegrationQuestion and Answers: Page 166

Question Number 93904 Answers: 1 Comments: 0

∫_0 ^∞ (e^(−ax) −e^(−bx) ) dx

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\left({e}^{−{ax}} −{e}^{−{bx}} \right)\:{dx}\: \\ $$

Question Number 93898 Answers: 1 Comments: 1

let f(x)=2(√(3−x)) and g(x) =x^2 −2x +5 1) calculate fog(x) and determine D_(fog) 2) calculate ∫ fog(x)dx 3) calculate ∫ ((f^(−1) (x))/(f(x)))dx and ∫ ((f^(−1) og(x))/(fog(x)))dx

$${let}\:{f}\left({x}\right)=\mathrm{2}\sqrt{\mathrm{3}−{x}}\:{and}\:{g}\left({x}\right)\:={x}^{\mathrm{2}} −\mathrm{2}{x}\:+\mathrm{5} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{fog}\left({x}\right)\:{and}\:{determine}\:{D}_{{fog}} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\:\int\:{fog}\left({x}\right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int\:\frac{{f}^{−\mathrm{1}} \left({x}\right)}{{f}\left({x}\right)}{dx}\:\:{and}\:\:\int\:\frac{{f}^{−\mathrm{1}} {og}\left({x}\right)}{{fog}\left({x}\right)}{dx} \\ $$

Question Number 93873 Answers: 1 Comments: 1

find ∫_(−∞) ^∞ (dx/((x^2 +2x+4)^3 ))

$${find}\:\int_{−\infty} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}\right)^{\mathrm{3}} } \\ $$

Question Number 93869 Answers: 0 Comments: 2

∫ (dx/(1−tan^2 x)) ?

$$\int\:\frac{\mathrm{dx}}{\mathrm{1}−\mathrm{tan}\:^{\mathrm{2}} \mathrm{x}}\:? \\ $$

Question Number 93860 Answers: 0 Comments: 5

Consider the progression (I_n )_(n∈N) where I_n =∫_0 ^1 ((sin(πt))/(t+n))dt 1\ Show that: ∀n≥0, 0≤I_(n+1) ≤I_n and deduce that the series is convergent. 2\ Show that 0≤I_n ≤ln(((n+1)/n)) and deduce the limit the series (I_n )_(n∈N) 3\ Calculate lim_(n→+∞) (nI_n )

$$\mathrm{Consider}\:\mathrm{the}\:\mathrm{progression}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \:\mathrm{where}\:\mathrm{I}_{\mathrm{n}} \:=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{sin}\left(\pi\mathrm{t}\right)}{\mathrm{t}+\mathrm{n}}\mathrm{dt} \\ $$$$\mathrm{1}\backslash\:\mathrm{Show}\:\mathrm{that}:\:\forall\mathrm{n}\geqslant\mathrm{0},\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}+\mathrm{1}} \leqslant\mathrm{I}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{deduce}\:\mathrm{that}\:\mathrm{the}\: \\ $$$$\mathrm{series}\:\mathrm{is}\:\mathrm{convergent}. \\ $$$$\mathrm{2}\backslash\:\mathrm{Show}\:\mathrm{that}\:\mathrm{0}\leqslant\mathrm{I}_{\mathrm{n}} \leqslant\mathrm{ln}\left(\frac{\mathrm{n}+\mathrm{1}}{\mathrm{n}}\right)\:\mathrm{and}\:\mathrm{deduce}\:\mathrm{the}\:\mathrm{limit} \\ $$$$\mathrm{the}\:\mathrm{series}\:\left(\mathrm{I}_{\mathrm{n}} \right)_{\mathrm{n}\in\mathbb{N}} \\ $$$$\mathrm{3}\backslash\:\mathrm{Calculate}\:\underset{\mathrm{n}\rightarrow+\infty} {\mathrm{lim}}\left(\mathrm{nI}_{\mathrm{n}} \right) \\ $$

Question Number 93820 Answers: 1 Comments: 1

∫ ((x^2 dx)/(√(x^2 −x+1)))

$$\int\:\frac{{x}^{\mathrm{2}} \:{dx}}{\sqrt{{x}^{\mathrm{2}} −{x}+\mathrm{1}}}\: \\ $$

Question Number 93804 Answers: 1 Comments: 3

∫ x (((3x−1)/(x+2)))^(1/(3 )) dx ?

$$\int\:{x}\:\sqrt[{\mathrm{3}\:\:}]{\frac{\mathrm{3}{x}−\mathrm{1}}{{x}+\mathrm{2}}}\:{dx}\:?\: \\ $$

Question Number 94574 Answers: 1 Comments: 1

∫((√(cosx))/((√(sinx )) +(√(cosx))))dx=?

$$\int\frac{\sqrt{\mathrm{cosx}}}{\sqrt{\mathrm{sinx}\:}\:+\sqrt{\mathrm{cosx}}}\mathrm{dx}=? \\ $$

Question Number 93762 Answers: 1 Comments: 0