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AllQuestion and Answers: Page 1630

Question Number 43187 Answers: 2 Comments: 1

Question Number 43159 Answers: 1 Comments: 0

Question Number 43158 Answers: 2 Comments: 1

∫cosecxdx

$$\int{cosecxdx} \\ $$

Question Number 43157 Answers: 1 Comments: 2

∫secxdx

$$\int{secxdx} \\ $$

Question Number 43156 Answers: 1 Comments: 0

integrate w.r.t x ∫((xe^x )/(√(1+x^2 )))dx

$${integrate}\:{w}.{r}.{t}\:{x} \\ $$$$\int\frac{{xe}^{{x}} }{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 43170 Answers: 1 Comments: 3

let A_n = ∫_0 ^∞ [n e^(−x) ]dx with n integr natural. 1) calculate A_n 2) find lim_(n→+∞) A_n .

$${let}\:{A}_{{n}} =\:\int_{\mathrm{0}} ^{\infty} \:\:\left[{n}\:{e}^{−{x}} \right]{dx}\:\:{with}\:{n}\:{integr}\:{natural}. \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{A}_{{n}} . \\ $$

Question Number 43147 Answers: 0 Comments: 2

If ∫_0 ^∞ e^(−x^2 ) dx = ((√π)/(2 )) , then prove that ∫_0 ^∞ e^(−ax^2 ) dx = (√(π/(4a))) where a>0.

$$\mathrm{If}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−{x}^{\mathrm{2}} } {dx}\:=\:\frac{\sqrt{\pi}}{\mathrm{2}\:}\:, \\ $$$$\mathrm{then}\:{prove}\:{that}\:\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{a}{x}^{\mathrm{2}} } {dx}\:=\:\sqrt{\frac{\pi}{\mathrm{4a}}} \\ $$$$\mathrm{where}\:\mathrm{a}>\mathrm{0}. \\ $$

Question Number 43145 Answers: 1 Comments: 4

∫_0 ^∞ [ 2e^(−x) ]dx = ? where [.]= gif.

$$\int_{\mathrm{0}} ^{\infty} \:\left[\:\mathrm{2e}^{−{x}} \right]{dx}\:=\:?\: \\ $$$${where}\:\left[.\right]=\:{gif}. \\ $$

Question Number 43135 Answers: 1 Comments: 4

Construct a triangle ΔABC with ∠B=50° AC=6 cm AB+BC=8 cm see also Q42942.

$${Construct}\:{a}\:{triangle}\:\Delta{ABC}\:{with} \\ $$$$\angle{B}=\mathrm{50}° \\ $$$${AC}=\mathrm{6}\:{cm} \\ $$$${AB}+{BC}=\mathrm{8}\:{cm} \\ $$$$ \\ $$$${see}\:{also}\:{Q}\mathrm{42942}. \\ $$

Question Number 43131 Answers: 2 Comments: 1

Question Number 43125 Answers: 1 Comments: 1

Question Number 43100 Answers: 1 Comments: 2

let f(x) = ∫_0 ^(π/2) ((cosθ)/(1+xsinθ))dθ 1) determine a explicit form of f(x) 2) calculate ∫_0 ^(π/2) ((sin(2θ))/((1+xsinθ)^2 ))dθ 3) find the values of ∫_0 ^(π/2) ((cosθ)/(1+2cosθ))dθ and ∫_0 ^(π/2) ((sin(2θ))/((1+3sinθ)^2 ))dθ .

$${let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{cos}\theta}{\mathrm{1}+{xsin}\theta}{d}\theta \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+{xsin}\theta\right)^{\mathrm{2}} }{d}\theta \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{values}\:{of}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\frac{{cos}\theta}{\mathrm{1}+\mathrm{2}{cos}\theta}{d}\theta\:\:\:{and}\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\:\:\:\:\frac{{sin}\left(\mathrm{2}\theta\right)}{\left(\mathrm{1}+\mathrm{3}{sin}\theta\right)^{\mathrm{2}} }{d}\theta\:. \\ $$

Question Number 43080 Answers: 0 Comments: 4