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IntegrationQuestion and Answers: Page 199

Question Number 83204 Answers: 0 Comments: 1

∫_(t−1) ^t ln(x!)dx=?

$$\int_{{t}−\mathrm{1}} ^{{t}} {ln}\left({x}!\right){dx}=? \\ $$

Question Number 83164 Answers: 2 Comments: 2

Evaluate: ∫_0 ^( (π/2)) (( 1)/(1+cos 𝛂 cos x))dx

$$ \\ $$$$ \\ $$$$\:\mathrm{Evaluate}: \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{\:\:\mathrm{1}}{\mathrm{1}+\boldsymbol{\mathrm{cos}}\:\boldsymbol{\alpha}\:\boldsymbol{\mathrm{cos}}\:\boldsymbol{\mathrm{x}}}\boldsymbol{\mathrm{dx}} \\ $$

Question Number 83123 Answers: 0 Comments: 6

∫(((x^2 −1))/(((√(x^2 +1)))(x^2 +2x−2))) dx

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

Question Number 83115 Answers: 1 Comments: 1

∫cos xe^(sin x) dx

$$\int\mathrm{cos}\:{xe}^{\mathrm{sin}\:{x}} {dx} \\ $$

Question Number 83110 Answers: 0 Comments: 10

bounded by the curve y=(√(4-x)) y=0 y=1

$${bounded}\:{by}\:{the}\:{curve}\:{y}=\sqrt{\mathrm{4}-{x}}\:{y}=\mathrm{0}\:{y}=\mathrm{1} \\ $$

Question Number 83109 Answers: 0 Comments: 1

∫_(1/e) ^e (dt/t)

$$\int_{\mathrm{1}/\boldsymbol{{e}}} ^{{e}} \frac{\boldsymbol{{dt}}}{\boldsymbol{{t}}} \\ $$

Question Number 83108 Answers: 1 Comments: 0

prove that ∫_0 ^(π/4) ((cos(nx))/(cos^n (x))) dx =2^n [(π/8)−Σ_(k=1) ^(n−1) ((sin(((kπ)/4)))/(2k((√2))^k ))] n∈N^∗

$${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \frac{{cos}\left({nx}\right)}{{cos}^{{n}} \left({x}\right)}\:{dx}\:=\mathrm{2}^{{n}} \left[\frac{\pi}{\mathrm{8}}−\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\sum}}\frac{{sin}\left(\frac{{k}\pi}{\mathrm{4}}\right)}{\mathrm{2}{k}\left(\sqrt{\mathrm{2}}\right)^{{k}} }\right]\:{n}\in{N}^{\ast} \\ $$

Question Number 83104 Answers: 1 Comments: 0

∫((e^x dx)/(3+e^x ))

$$\int\frac{{e}^{{x}} {dx}}{\mathrm{3}+{e}^{{x}} } \\ $$

Question Number 83096 Answers: 0 Comments: 1

∫tan x^4 dx

$$\int\mathrm{tan}\:{x}^{\mathrm{4}} {dx} \\ $$

Question Number 83095 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83094 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83093 Answers: 0 Comments: 1

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83092 Answers: 0 Comments: 0

∫cosec x^5 dx

$$\int\mathrm{cosec}\:{x}^{\mathrm{5}} {dx} \\ $$

Question Number 83085 Answers: 1 Comments: 2

1) find ∫ (dx/((x^2 +1)^4 )) 2)calculate ∫_0 ^∞ (dx/((x^2 +1)^4 ))

$$\left.\mathrm{1}\right)\:{find}\:\int\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{4}} } \\ $$

Question Number 83064 Answers: 2 Comments: 0

show that ∫_0 ^(π/2) ((sin(nx))/(sin(x)))dx=(π/2) n is posative odd number

$${show}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{sin}\left({nx}\right)}{{sin}\left({x}\right)}{dx}=\frac{\pi}{\mathrm{2}} \\ $$$${n}\:{is}\:{posative}\:{odd}\:{number} \\ $$

Question Number 83063 Answers: 1 Comments: 3

Question Number 83010 Answers: 0 Comments: 0

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

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

Question Number 83009 Answers: 0 Comments: 1

calculate ∫_0 ^∞ ((cos(chx))/(x^2 +3))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{cos}\left({chx}\right)}{{x}^{\mathrm{2}} \:+\mathrm{3}}{dx} \\ $$

Question Number 83008 Answers: 0 Comments: 0

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

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

Question Number 82974 Answers: 1 Comments: 1

Question Number 82972 Answers: 0 Comments: 0

calculate ∫_0 ^1 arctan(x)ln(1+x)dx

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

Question Number 82971 Answers: 1 Comments: 2

1)find ∫ (dx/((x^2 +x+1)^6 )) 2)calculate ∫_(−∞) ^(+∞) (dx/((x^2 +x+1)^6 ))

$$\left.\mathrm{1}\right){find}\:\int\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{6}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{−\infty} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{1}\right)^{\mathrm{6}} } \\ $$

Question Number 82970 Answers: 0 Comments: 2

1)find ∫ (dx/((x^2 −1)^9 )) 2) calculate ∫_2 ^(+∞) (dx/((x^2 −1)^9 ))

$$\left.\mathrm{1}\right){find}\:\int\:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{9}} } \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{2}} ^{+\infty} \:\frac{{dx}}{\left({x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{9}} } \\ $$

Question Number 82954 Answers: 1 Comments: 3

∫ ((cos 4x−cos 2x)/(sin 4x−cos 2x)) dx

$$\int\:\frac{\mathrm{cos}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}{\mathrm{sin}\:\mathrm{4x}−\mathrm{cos}\:\mathrm{2x}}\:\mathrm{dx}\: \\ $$

Question Number 82891 Answers: 2 Comments: 0

∫ (1/(√(1−x^4 ))) dx = ?

$$\int\:\frac{\mathrm{1}}{\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{4}} }}\:\mathrm{dx}\:=\:? \\ $$

Question Number 82872 Answers: 0 Comments: 1

1)find ∫∫_W ((xdx)/(a^2 +x^2 +y^2 )) with W_a →x^2 +y^2 ≤a^2 and x>0 (a>0) 2)calculate ∫∫_W_1 ((xdx)/(x^2 +y^2 +1))

$$\left.\mathrm{1}\right){find}\:\int\int_{{W}} \:\frac{{xdx}}{{a}^{\mathrm{2}} \:+{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:{with} \\ $$$${W}_{{a}} \rightarrow{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \:\leqslant{a}^{\mathrm{2}} \:{and}\:{x}>\mathrm{0}\:\:\:\:\:\left({a}>\mathrm{0}\right) \\ $$$$\left.\mathrm{2}\right){calculate}\:\int\int_{{W}_{\mathrm{1}} } \:\:\:\frac{{xdx}}{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \:+\mathrm{1}} \\ $$

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