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IntegrationQuestion and Answers: Page 51

Question Number 168311 Answers: 0 Comments: 5

Calculate∫(((arcsin(x))^2 )/(1+x^2 ))dx

$${Calculate}\int\frac{\left({arcsin}\left({x}\right)\right)^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 168303 Answers: 2 Comments: 0

calculate ∫_0 ^1 x(√(1−x^6 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}\sqrt{\mathrm{1}−{x}^{\mathrm{6}} }{dx} \\ $$

Question Number 168296 Answers: 2 Comments: 0

∫(5x+2)cos(2x)dx=?

$$\int\left(\mathrm{5}{x}+\mathrm{2}\right){cos}\left(\mathrm{2}{x}\right){dx}=? \\ $$

Question Number 168188 Answers: 4 Comments: 1

∫ ((1−sin 2x)/((1+sin 2x)^2 )) dx =?

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

Question Number 167989 Answers: 2 Comments: 0

∫ ((3−cos x)/(3+cos x)) dx =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\frac{\mathrm{3}−\mathrm{cos}\:{x}}{\mathrm{3}+\mathrm{cos}\:{x}}\:{dx}\:=? \\ $$

Question Number 167965 Answers: 1 Comments: 0

Question Number 167922 Answers: 4 Comments: 2

Find the value of Q if ∫_0 ^Q (√(cosec θ−1)) dθ=ln (((3+2(√2))/2))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{Q}\:\:\mathrm{if} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{Q}} \sqrt{\mathrm{cosec}\:\theta−\mathrm{1}}\:\mathrm{d}\theta=\mathrm{ln}\:\left(\frac{\mathrm{3}+\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{2}}\right) \\ $$$$ \\ $$

Question Number 167912 Answers: 1 Comments: 0

∫^(π/4) _0 ((cos(12x))/(cos^(14) (x)))dx=?

$$\underset{\mathrm{0}} {\int}^{\frac{\pi}{\mathrm{4}}} \frac{{cos}\left(\mathrm{12}{x}\right)}{{cos}^{\mathrm{14}} \left({x}\right)}{dx}=? \\ $$

Question Number 167885 Answers: 0 Comments: 0

Question Number 167862 Answers: 1 Comments: 0

■Nice integral■ 𝛀=∫_0 ^1 ((ln^3 (1−x^2 ))/( (√(1−x^2 ))))dx ? by MATH.AMIN

$$\blacksquare\boldsymbol{\mathrm{Nice}}\:\boldsymbol{\mathrm{integral}}\blacksquare \\ $$$$\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\boldsymbol{\mathrm{ln}}^{\mathrm{3}} \left(\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} \right)}{\:\sqrt{\mathrm{1}−\boldsymbol{\mathrm{x}}^{\mathrm{2}} }}\boldsymbol{\mathrm{dx}}\:\:\:? \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{MATH}}.\boldsymbol{\mathrm{AMIN}} \\ $$

Question Number 167823 Answers: 1 Comments: 2

Question Number 167716 Answers: 1 Comments: 0

Question Number 167713 Answers: 1 Comments: 0

Question Number 167647 Answers: 2 Comments: 0

Question Number 167626 Answers: 0 Comments: 0

∫_0 ^∞ (e^(−t) /( (√t))) e^(−(1/(4t))) dt

$$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{t}} }{\:\sqrt{{t}}}\:{e}^{−\frac{\mathrm{1}}{\mathrm{4}{t}}} \:{dt} \\ $$

Question Number 167598 Answers: 1 Comments: 0

∫ (x/(5x^4 +x^3 −5x−1)) dx=?

$$\:\:\:\:\:\:\int\:\frac{\mathrm{x}}{\mathrm{5x}^{\mathrm{4}} +\mathrm{x}^{\mathrm{3}} −\mathrm{5x}−\mathrm{1}}\:\mathrm{dx}=? \\ $$

Question Number 167596 Answers: 1 Comments: 0

∫ ((cos 7x−cos 8x)/(1+2cos 5x)) dx =?

$$\:\:\int\:\frac{\mathrm{cos}\:\mathrm{7x}−\mathrm{cos}\:\mathrm{8x}}{\mathrm{1}+\mathrm{2cos}\:\mathrm{5x}}\:\mathrm{dx}\:=? \\ $$

Question Number 167586 Answers: 2 Comments: 1

λ=∫ (dx/( (√(1+cos x))+(√(1+sin x)))) =?

$$\:\:\:\:\:\:\:\:\lambda=\int\:\frac{\mathrm{dx}}{\:\sqrt{\mathrm{1}+\mathrm{cos}\:\mathrm{x}}+\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}\:=? \\ $$

Question Number 167562 Answers: 1 Comments: 0

∫ (e^(−x) /(1+e^x )) dx=?

$$\:\:\:\:\:\:\:\:\int\:\frac{{e}^{−{x}} }{\mathrm{1}+{e}^{{x}} }\:{dx}=? \\ $$

Question Number 167510 Answers: 1 Comments: 0

explicite f(a)=∫_0 ^(π/2) ln(a+tan^2 x)dx a≥2

$${explicite}\:{f}\left({a}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {ln}\left({a}+{tan}^{\mathrm{2}} {x}\right){dx} \\ $$$${a}\geqslant\mathrm{2} \\ $$

Question Number 167644 Answers: 1 Comments: 2

find the exact value of ∫_0 ^(2π) (dx/( (√(1+sin x))+(√(1+cos x))))

$$\mathrm{find}\:\mathrm{the}\:\mathrm{exact}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\frac{{dx}}{\:\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}+\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}} \\ $$

Question Number 167438 Answers: 2 Comments: 0

∫ ((sin 4x)/(4sin^4 x−4sin^2 x+1)) dx=?

$$\:\:\int\:\frac{\mathrm{sin}\:\mathrm{4x}}{\mathrm{4sin}\:^{\mathrm{4}} \mathrm{x}−\mathrm{4sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{1}}\:\mathrm{dx}=? \\ $$

Question Number 167433 Answers: 0 Comments: 0

calculate 1 : Ω= Σ_(n=1) ^∞ ((( H_( n) )/n) )^( 2) = ? 2 : ∫_0 ^( 1) ((ln^( 2) (x).Li_( 2) (x))/x) dx = ?

$$ \\ $$$$\:\:\:\:\:\:\:{calculate}\: \\ $$$$\:\:\:\:\mathrm{1}\::\:\:\Omega=\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left(\frac{\:{H}_{\:{n}} }{{n}}\:\right)^{\:\mathrm{2}} =\:? \\ $$$$\:\:\:\:\mathrm{2}\::\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{ln}^{\:\mathrm{2}} \left({x}\right).\mathrm{L}{i}_{\:\mathrm{2}} \left({x}\right)}{{x}}\:{dx}\:=\:? \\ $$$$ \\ $$

Question Number 167416 Answers: 0 Comments: 3

∫_( 0) ^( π) ((x.sin (x))/(1+cos^2 (x) ))dx ∫_0 ^( π) (1/(1+cos^3 (x) +sin^3 (x) ))dx ∫_0 ^(π/2) ((x.cos(x)sin(x))/(tan^2 (x)+ cot^2 (x) ))dx ∫_0 ^( 1) tan^(−1) ((√(1−x^2 )))dx

$$\int_{\:\mathrm{0}} ^{\:\pi} \:\frac{{x}.\mathrm{sin}\:\left({x}\right)}{\mathrm{1}+\mathrm{cos}^{\mathrm{2}} \left({x}\right)\:}{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\:\:\pi} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{cos}^{\mathrm{3}} \left({x}\right)\:+\mathrm{sin}^{\mathrm{3}} \left({x}\right)\:}{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{x}.\mathrm{cos}\left(\mathrm{x}\right)\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{tan}^{\mathrm{2}} \left(\mathrm{x}\right)+\:\mathrm{cot}^{\mathrm{2}} \left(\mathrm{x}\right)\:}\mathrm{dx} \\ $$$$ \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \mathrm{tan}^{−\mathrm{1}} \left(\sqrt{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\right)\mathrm{dx} \\ $$$$ \\ $$

Question Number 167400 Answers: 2 Comments: 0

Question Number 167366 Answers: 0 Comments: 0

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