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IntegrationQuestion and Answers: Page 59

Question Number 161652 Answers: 0 Comments: 0

∫_0 ^( 1) ((xlog(a+x))/(1+x^2 ))dx ∀ ∣a∣ ∈ N

$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{xlog}\left(\mathrm{a}+\mathrm{x}\right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\:\forall\:\mid\mathrm{a}\mid\:\in\:\mathbb{N} \\ $$

Question Number 161703 Answers: 2 Comments: 1

(1)∫ ((sin x−cos x)/( (√(sin 2x)))) dx (2) ∫_0 ^( π/2) cos 7x cos 17x cos 37x dx

$$\left(\mathrm{1}\right)\int\:\frac{\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{sin}\:\mathrm{2}{x}}}\:{dx} \\ $$$$\left(\mathrm{2}\right)\:\int_{\mathrm{0}} ^{\:\pi/\mathrm{2}} \mathrm{cos}\:\mathrm{7}{x}\:\mathrm{cos}\:\mathrm{17}{x}\:\mathrm{cos}\:\mathrm{37}{x}\:{dx} \\ $$

Question Number 161646 Answers: 0 Comments: 0

Question Number 161609 Answers: 0 Comments: 0

∫_0 ^1 ((xln(1+x^4 ))/(1+x^2 ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{{xln}}\left(\mathrm{1}+\boldsymbol{{x}}^{\mathrm{4}} \right)}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} }\boldsymbol{{dx}}=? \\ $$$$ \\ $$

Question Number 161537 Answers: 2 Comments: 0

∫_0 ^( (π/4)) ((1+tan^4 (x))/(cot^2 (x))) dx =?

$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \:\frac{\mathrm{1}+\mathrm{tan}\:^{\mathrm{4}} \left({x}\right)}{\mathrm{cot}\:^{\mathrm{2}} \left({x}\right)}\:{dx}\:=? \\ $$

Question Number 161443 Answers: 1 Comments: 0

Question Number 161412 Answers: 0 Comments: 0

Question Number 161407 Answers: 1 Comments: 0

Question Number 161404 Answers: 1 Comments: 0

Question Number 161393 Answers: 0 Comments: 0

Question Number 161329 Answers: 0 Comments: 0

∫_1 ^( 2) ((tan^(−1) (x−1)log(x))/x)dx

$$\int_{\mathrm{1}} ^{\:\mathrm{2}} \frac{\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{x}−\mathrm{1}\right)\mathrm{log}\left(\mathrm{x}\right)}{\mathrm{x}}\mathrm{dx} \\ $$

Question Number 161285 Answers: 5 Comments: 0

(1) ∫ (dx/(1−2cos x)) (2) ∫ ((sin 2x)/(sin x−sin^2 2x)) dx (3) ∫ (dx/(cos 2x−sin x))

$$\left(\mathrm{1}\right)\:\int\:\frac{{dx}}{\mathrm{1}−\mathrm{2cos}\:{x}} \\ $$$$\left(\mathrm{2}\right)\:\int\:\frac{\mathrm{sin}\:\mathrm{2}{x}}{\mathrm{sin}\:{x}−\mathrm{sin}\:^{\mathrm{2}} \mathrm{2}{x}}\:{dx} \\ $$$$\left(\mathrm{3}\right)\:\int\:\frac{{dx}}{\mathrm{cos}\:\mathrm{2}{x}−\mathrm{sin}\:{x}} \\ $$

Question Number 161281 Answers: 0 Comments: 0

Question Number 161265 Answers: 1 Comments: 2

Question Number 161256 Answers: 1 Comments: 0

Given f(x)=f(x+2), ∀x∈R If ∫_0 ^2 f(x)dx= p then ∫_0 ^(2020) f(x+2a)dx=? for a∈Z^+

$$\:{Given}\:{f}\left({x}\right)={f}\left({x}+\mathrm{2}\right),\:\forall{x}\in\mathbb{R} \\ $$$$\:{If}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}=\:{p}\:{then}\:\underset{\mathrm{0}} {\overset{\mathrm{2020}} {\int}}{f}\left({x}+\mathrm{2}{a}\right){dx}=? \\ $$$$\:{for}\:{a}\in\mathbb{Z}^{+} \\ $$

Question Number 161233 Answers: 0 Comments: 0

Question Number 161229 Answers: 1 Comments: 0

Given f(x)= { ((1−∣x∣ ; x≤1)),((∣x∣−1 ; x>1)) :} find ∫_(−3) ^( 8) [f(x−1)+f(x+1)] dx.

$$\:{Given}\:{f}\left({x}\right)=\:\begin{cases}{\mathrm{1}−\mid{x}\mid\:;\:{x}\leqslant\mathrm{1}}\\{\mid{x}\mid−\mathrm{1}\:;\:{x}>\mathrm{1}}\end{cases} \\ $$$$\:{find}\:\int_{−\mathrm{3}} ^{\:\mathrm{8}} \left[{f}\left({x}−\mathrm{1}\right)+{f}\left({x}+\mathrm{1}\right)\right]\:{dx}.\: \\ $$

Question Number 161212 Answers: 2 Comments: 2

∫_( 0) ^( (π/2)) ((x sin x cos x)/(cos^4 x +sin^4 x)) dx =?

$$\:\:\int_{\:\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\frac{{x}\:\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}{\mathrm{cos}\:^{\mathrm{4}} {x}\:+\mathrm{sin}\:^{\mathrm{4}} {x}}\:{dx}\:=? \\ $$

Question Number 161178 Answers: 1 Comments: 0

∫^∞ _2 ((arctg(x))/(arctg((x/2))))dx=???

$$\underset{\mathrm{2}} {\int}^{\infty} \frac{\boldsymbol{{arctg}}\left(\boldsymbol{{x}}\right)}{\boldsymbol{{arctg}}\left(\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)}\boldsymbol{{dx}}=??? \\ $$

Question Number 161176 Answers: 0 Comments: 0

calculate Θ := Σ_(n=1) ^∞ (( (−1 )^( n−1) )/(n ( n + (1/3) ))) =? ■ m.n −−−−−−−−−−−−−

$$ \\ $$$$\:\:\:\:\:\:{calculate}\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\Theta\::=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\:\right)^{\:{n}−\mathrm{1}} }{{n}\:\left(\:{n}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\:\right)}\:=?\:\:\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:{m}.{n} \\ $$$$\:\:\:\:\:\:\:−−−−−−−−−−−−− \\ $$$$ \\ $$

Question Number 161100 Answers: 0 Comments: 0

f(x^2 )= 2+∫_( 0) ^( x^2 ) f(y) (1−tan y)dy , ∀x∈R f(−π)=?

$$\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)=\:\mathrm{2}+\int_{\:\mathrm{0}} ^{\:\mathrm{x}^{\mathrm{2}} } \mathrm{f}\left(\mathrm{y}\right)\:\left(\mathrm{1}−\mathrm{tan}\:\mathrm{y}\right)\mathrm{dy}\:,\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{f}\left(−\pi\right)=? \\ $$

Question Number 161089 Answers: 3 Comments: 0

prove that I= ∫_0 ^( (π/2)) ln ( 1+ sin (2 α )) dα = 2G − π ln ((√2) ) G: catalan constant

$$ \\ $$$$\:\:{prove}\:{that} \\ $$$$\:\:\:\mathrm{I}=\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \mathrm{ln}\:\left(\:\mathrm{1}+\:{sin}\:\left(\mathrm{2}\:\alpha\:\right)\right)\:{d}\alpha\: \\ $$$$\:\:\:\:\:\:\:\:\:\:=\:\:\mathrm{2G}\:−\:\pi\:\mathrm{ln}\:\left(\sqrt{\mathrm{2}}\:\right) \\ $$$$\:\:\:\:\:\:\:\mathrm{G}:\:\:{catalan}\:{constant} \\ $$

Question Number 161076 Answers: 1 Comments: 0

Ω = ∫_0 ^( ∞) ((ln (1+ x ))/((1+ x^( 2) )^( 2) )) dx = ? −−−−−−−−−−−−

$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\:\left(\mathrm{1}+\:{x}\:\right)}{\left(\mathrm{1}+\:{x}^{\:\mathrm{2}} \right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:−−−−−−−−−−−− \\ $$$$\:\:\:\:\:\:\:\: \\ $$

Question Number 161003 Answers: 0 Comments: 0

Question Number 160982 Answers: 0 Comments: 0

Ω = ∫_0 ^( 1) (( ln (−ln (x)))/(1+x)) dx =^? ((−1)/2) ln^( 2) (2)

$$ \\ $$$$\:\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:{ln}\:\left(−{ln}\:\left({x}\right)\right)}{\mathrm{1}+{x}}\:{dx}\:\overset{?} {=}\frac{−\mathrm{1}}{\mathrm{2}}\:{ln}^{\:\mathrm{2}} \left(\mathrm{2}\right) \\ $$

Question Number 160979 Answers: 1 Comments: 0

Ω=∫_0 ^1 x^(n−1) ln(1−x)dx=??? n≥1

$$\Omega=\int_{\mathrm{0}} ^{\mathrm{1}} \boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{n}}−\mathrm{1}} \boldsymbol{\mathrm{ln}}\left(\mathrm{1}−\boldsymbol{\mathrm{x}}\right)\boldsymbol{\mathrm{dx}}=???\:\:\: \\ $$$$\boldsymbol{\mathrm{n}}\geqslant\mathrm{1} \\ $$

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