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IntegrationQuestion and Answers: Page 294

Question Number 36541 Answers: 0 Comments: 1

∫(x+(√(1+x^2 )) )^n dx

$$\int\left({x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\right)^{{n}} {dx} \\ $$

Question Number 36538 Answers: 0 Comments: 1

∫(dx/((a+bx^2 )(√(b−ax^2 ))))

$$\int\frac{{dx}}{\left({a}+{bx}^{\mathrm{2}} \right)\sqrt{{b}−{ax}^{\mathrm{2}} \:}}\:\: \\ $$

Question Number 36537 Answers: 0 Comments: 3

Question Number 36536 Answers: 0 Comments: 0

Question Number 36535 Answers: 0 Comments: 0

∫(dx/(secx+cosecx))

$$\int\frac{{dx}}{{secx}+{cosecx}} \\ $$

Question Number 36534 Answers: 0 Comments: 0

Question Number 36533 Answers: 0 Comments: 0

Question Number 36491 Answers: 2 Comments: 0

Question Number 36459 Answers: 2 Comments: 0

∫ e^(tan θ) (sec θ −sin θ) dθ = ?

$$\int\:\mathrm{e}^{\mathrm{tan}\:\theta} \left(\mathrm{sec}\:\theta\:−\mathrm{sin}\:\theta\right)\:\mathrm{d}\theta\:=\:? \\ $$

Question Number 36444 Answers: 0 Comments: 11

∫ (dx/((2x−7)(√((x−3)(x−4))))) = ?

$$\int\:\frac{\mathrm{d}{x}}{\left(\mathrm{2}{x}−\mathrm{7}\right)\sqrt{\left({x}−\mathrm{3}\right)\left({x}−\mathrm{4}\right)}}\:=\:? \\ $$

Question Number 36441 Answers: 1 Comments: 1

find ∫ (e^(tanx) /(cos^2 x))dx

$${find}\:\:\int\:\:\:\frac{{e}^{{tanx}} }{{cos}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 36440 Answers: 0 Comments: 0

find ∫ (dx/((3+x^2 )^(1/3) ))

$${find}\:\int\:\:\:\:\frac{{dx}}{\left(\mathrm{3}+{x}^{\mathrm{2}} \right)^{\frac{\mathrm{1}}{\mathrm{3}}} } \\ $$

Question Number 36439 Answers: 1 Comments: 1

calculate ∫_0 ^π ((sin(2x))/(2 +cosx))dx

$${calculate}\:\int_{\mathrm{0}} ^{\pi} \:\:\:\frac{{sin}\left(\mathrm{2}{x}\right)}{\mathrm{2}\:+{cosx}}{dx} \\ $$

Question Number 36438 Answers: 1 Comments: 3

let F(x) =∫_x ^(1/x) ((arctan(t))/t)dt 1) calculate (dF/dx)(x) 2) find F(x).

$${let}\:{F}\left({x}\right)\:=\int_{{x}} ^{\frac{\mathrm{1}}{{x}}} \:\:\frac{{arctan}\left({t}\right)}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:\frac{{dF}}{{dx}}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{F}\left({x}\right). \\ $$

Question Number 36436 Answers: 2 Comments: 0

find ∫ ((sinx)/(1+cos^3 x))dx

$${find}\:\:\int\:\:\:\:\:\frac{{sinx}}{\mathrm{1}+{cos}^{\mathrm{3}} {x}}{dx} \\ $$

Question Number 36435 Answers: 0 Comments: 4

find the value of h(t)=∫_0 ^1 ln(1+tx^2 ) with ∣t∣≤1 2) calculate ∫_0 ^1 ln(1+x^2 )dx 3) calculate ∫_0 ^1 ln(1−x^2 )dx

$${find}\:{the}\:{value}\:{of}\:{h}\left({t}\right)=\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}+{tx}^{\mathrm{2}} \right)\:\:{with}\:\mid{t}\mid\leqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right){dx} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right){dx} \\ $$

Question Number 36434 Answers: 1 Comments: 1

find g(x) =∫_0 ^x (e^(−t) /(√(1+t^2 )))dt.

$${find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{{x}} \:\:\:\frac{{e}^{−{t}} }{\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}{dt}. \\ $$

Question Number 36433 Answers: 2 Comments: 1

valculate f(x)= ∫_0 ^2 (((x+2)^2 )/(√(x^2 +4x+5)))dx

$${valculate}\:{f}\left({x}\right)=\:\int_{\mathrm{0}} ^{\mathrm{2}} \:\:\frac{\left({x}+\mathrm{2}\right)^{\mathrm{2}} }{\sqrt{{x}^{\mathrm{2}} \:+\mathrm{4}{x}+\mathrm{5}}}{dx} \\ $$

Question Number 36432 Answers: 1 Comments: 1

calculate I = ∫_(−∞) ^(+∞) ((x+1)/((x^2 +1)^2 ))dx .

$${calculate}\:\:{I}\:=\:\int_{−\infty} ^{+\infty} \:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx}\:. \\ $$

Question Number 36431 Answers: 0 Comments: 1

calculate ∫_0 ^1 ((x+1)/((x^2 +1)^2 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}+\mathrm{1}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 36430 Answers: 1 Comments: 1

find ∫ ((ln(x+x^2 ))/x^2 )dx

$${find}\:\:\int\:\:\:\frac{{ln}\left({x}+{x}^{\mathrm{2}} \right)}{{x}^{\mathrm{2}} }{dx}\: \\ $$

Question Number 36429 Answers: 1 Comments: 1

let ϕ(λ) = ∫_(λ/π) ^(π/λ) (1+(1/x^2 ))arctan(x)dx with λ>0 1) find a simple form of ϕ(λ) 2) calculate ϕ^′ (λ).

$${let}\:\:\varphi\left(\lambda\right)\:=\:\int_{\frac{\lambda}{\pi}} ^{\frac{\pi}{\lambda}} \left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right){arctan}\left({x}\right){dx}\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{a}\:{simple}\:{form}\:{of}\:\varphi\left(\lambda\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\varphi^{'} \left(\lambda\right). \\ $$

Question Number 36428 Answers: 2 Comments: 1

find ∫ (dx/(cos^4 x +sin^4 x))

$${find}\:\:\int\:\:\:\:\frac{{dx}}{{cos}^{\mathrm{4}} {x}\:+{sin}^{\mathrm{4}} {x}} \\ $$

Question Number 36427 Answers: 1 Comments: 1

calculate ∫_(π/8) ^(π/6) (dx/(sin(2x)))

$${calculate}\:\int_{\frac{\pi}{\mathrm{8}}} ^{\frac{\pi}{\mathrm{6}}} \:\:\:\frac{{dx}}{{sin}\left(\mathrm{2}{x}\right)} \\ $$

Question Number 36426 Answers: 1 Comments: 0

find ∫ x^2 (√(1+x^3 )) dx

$${find}\:\:\int\:{x}^{\mathrm{2}} \sqrt{\mathrm{1}+{x}^{\mathrm{3}} \:}\:{dx} \\ $$

Question Number 36425 Answers: 2 Comments: 2

calculate ∫_1 ^4 ((x(√x))/(x^2 −5x +4))dx

$${calculate}\:\int_{\mathrm{1}} ^{\mathrm{4}} \:\:\:\frac{{x}\sqrt{{x}}}{{x}^{\mathrm{2}} \:−\mathrm{5}{x}\:+\mathrm{4}}{dx} \\ $$

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