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IntegrationQuestion and Answers: Page 166

Question Number 93720 Answers: 0 Comments: 2

∫ (dx/((x+1)^3 (√(x^2 +2x))))

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

Question Number 93484 Answers: 0 Comments: 2

∫t^2 /(1+t^2 )^2 dx=

$$\int\mathrm{t}^{\mathrm{2}} /\left(\mathrm{1}+\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{2}} \mathrm{dx}= \\ $$

Question Number 93481 Answers: 1 Comments: 1

∫(log x/x^2 )dx=

$$\int\left(\mathrm{log}\:\mathrm{x}/\mathrm{x}^{\mathrm{2}} \right)\mathrm{dx}= \\ $$

Question Number 93473 Answers: 1 Comments: 1

∫1/(1+x^2 )^2

$$\int\mathrm{1}/\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} \\ $$

Question Number 93471 Answers: 1 Comments: 0

∫1/1+x2

$$\int\mathrm{1}/\mathrm{1}+\mathrm{x2} \\ $$

Question Number 93467 Answers: 1 Comments: 0

∫ (sin x+2cos x)^3 dx

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

Question Number 93410 Answers: 0 Comments: 7

∫((1+x^6 )/(1+x^8 ))dx

$$\int\frac{\mathrm{1}+{x}^{\mathrm{6}} }{\mathrm{1}+{x}^{\mathrm{8}} }{dx} \\ $$

Question Number 93265 Answers: 1 Comments: 1

calculate ∫_0 ^((2π)/3) (dx/(3+sin(3x)))

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

Question Number 93248 Answers: 1 Comments: 1

∫ _0 ^1 ln(x) dx

$$\int\underset{\mathrm{0}} {\overset{\mathrm{1}} {\:}}\:\mathrm{ln}\left(\mathrm{x}\right)\:\mathrm{dx}\: \\ $$

Question Number 93227 Answers: 0 Comments: 4

Question Number 93225 Answers: 0 Comments: 2

Question Number 93193 Answers: 1 Comments: 1

∫ ((ln(x+(√(x^2 −1))))/(√((x^2 −1)^3 ))) dx ?

$$\int\:\frac{\mathrm{ln}\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\right)}{\sqrt{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} }}\:\mathrm{dx}\:?\: \\ $$

Question Number 93175 Answers: 3 Comments: 1

∫ (dx/(√(sin^3 (x).cos^5 (x)))) ?

$$\int\:\frac{\mathrm{dx}}{\sqrt{\mathrm{sin}\:^{\mathrm{3}} \:\left(\mathrm{x}\right).\mathrm{cos}\:^{\mathrm{5}} \left(\mathrm{x}\right)}}\:?\: \\ $$

Question Number 93144 Answers: 1 Comments: 0

∫ ((x^3 −1)/(x^3 +6x^2 +10x)) dx

$$\int\:\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{1}}{\mathrm{x}^{\mathrm{3}} +\mathrm{6x}^{\mathrm{2}} +\mathrm{10x}}\:\mathrm{dx}\: \\ $$

Question Number 93109 Answers: 0 Comments: 2

∫_0 ^(2π) cos^(2020) (x)dx

$$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\:{cos}^{\mathrm{2020}} \left({x}\right){dx} \\ $$

Question Number 93098 Answers: 0 Comments: 5

find the general solution to ∫ (1/(a sin x + b cos x)) dx and ∫ (1/(a cos x − bsin x)) dx where a , b are constants.

$$\mathrm{find}\:\mathrm{the}\:\mathrm{general}\:\mathrm{solution}\:\mathrm{to} \\ $$$$\:\int\:\frac{\mathrm{1}}{{a}\:\mathrm{sin}\:{x}\:+\:{b}\:\mathrm{cos}\:{x}}\:{dx}\:\:\mathrm{and}\:\int\:\frac{\mathrm{1}}{{a}\:\mathrm{cos}\:{x}\:−\:{b}\mathrm{sin}\:{x}}\:{dx} \\ $$$$\mathrm{where}\:{a}\:,\:{b}\:\mathrm{are}\:\mathrm{constants}. \\ $$$$ \\ $$

Question Number 93061 Answers: 0 Comments: 3

∫_1 ^(+∞) ((sin u)/u)du

$$\int_{\mathrm{1}} ^{+\infty} \frac{\mathrm{sin}\:\mathrm{u}}{\mathrm{u}}\mathrm{du} \\ $$

Question Number 93045 Answers: 1 Comments: 2

∫ sin^(−1) ((√(x/2))) dx

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

Question Number 93038 Answers: 1 Comments: 1

∫_1 ^5 (√(x^3 +1)) dx = ?

$$\underset{\mathrm{1}} {\overset{\mathrm{5}} {\int}}\:\sqrt{\mathrm{x}^{\mathrm{3}} +\mathrm{1}}\:\mathrm{dx}\:=\:? \\ $$

Question Number 93031 Answers: 1 Comments: 11

Question Number 93005 Answers: 1 Comments: 0

∫ sec x (√(sec x+tan x)) dx =

$$\int\:\mathrm{sec}\:\mathrm{x}\:\sqrt{\mathrm{sec}\:\mathrm{x}+\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\:=\: \\ $$

Question Number 92993 Answers: 0 Comments: 1

calculate ∫_(−∞) ^(+∞) ((cos(cosx−sinx))/(x^2 +4))dx

$${calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{{cos}\left({cosx}−{sinx}\right)}{{x}^{\mathrm{2}} \:+\mathrm{4}}{dx} \\ $$

Question Number 93004 Answers: 0 Comments: 0

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

$$\int\:\frac{\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:\left(\mathrm{x}\right)\right)}{\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}\right)+\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)}\:\mathrm{dx}\:? \\ $$

Question Number 92976 Answers: 1 Comments: 0

define Δ_n =((n(1+n))/2) find a closer form for Σ_(n=1) ^∞ (1/Δ_n ^m )

$${define} \\ $$$$\Delta_{{n}} =\frac{{n}\left(\mathrm{1}+{n}\right)}{\mathrm{2}} \\ $$$${find}\:{a}\:{closer}\:{form}\:{for} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\Delta_{{n}} ^{{m}} } \\ $$

Question Number 92971 Answers: 0 Comments: 3

1) decompose F(x) =(1/(x^4 (x−3)^5 )) 2)calculate ∫_5 ^(+∞) (dx/(x^4 (x−3)^5 ))

$$\left.\mathrm{1}\right)\:{decompose}\:{F}\left({x}\right)\:=\frac{\mathrm{1}}{{x}^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{5}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{4}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 92960 Answers: 2 Comments: 0

∫ ((sin^2 x dx)/((1+cos x)^2 ))

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

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