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.......Advanced ...∗∗∗∗∗ ...Integral...... Prove that :: Φ :=∫_0 ^( 1) ((1−x)/((1−x+x^2 )log(x)))dx= proof:: Φ:=∫_0 ^( 1) ((1−x^2 )/((1−x^3 )log(x)))dx f (a):= ∫_0 ^( 1) ((1−x^a )/((1−x^3 )log(x))) Φ := f (2) ........✓ f ′(a):=∫_0 ^( 1) ((−x^a log(x))/((1−x^3 )log(x)))=∫_0 ^( 1) ((−x^a )/(1−x^3 ))dx (★) (★):: x^3 =y ⇒ f ′(a):=(1/3)∫_0 ^( 1) ((y^((−2)/3) −y^((a/3)−(2/3)) )/(1−y))dy :=(1/3)∫_0 ^( 1) ((y^((−2)/3) −1+1−y^((a/3)−(1/3)) )/(1−y))dy :=(1/3)(ψ((a/3)+(2/3))−ψ((2/3))) f (a):=log(Γ((a/3)+(2/3)))−(a/3)ψ((2/3))+C f (0):=0=log(Γ((2/3)))+C C :=−log(Γ((2/3))) Φ:= f (2)=log(Γ((4/3)))−(2/3) ψ((2/3))−log(Γ((2/3))) :=log(((Γ((4/3)))/(Γ((2/3)))))−(2/3)ψ((2/3)) ....✓

Question Number 142273 Answers: 0 Comments: 1

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

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

Question Number 142164 Answers: 0 Comments: 1

Question Number 142151 Answers: 0 Comments: 1

lim_(x→∞) sin (√(x+1))−sin (√x) =?

$$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{sin}\:\sqrt{{x}+\mathrm{1}}−\mathrm{sin}\:\sqrt{{x}}\:=? \\ $$

Question Number 142150 Answers: 1 Comments: 1

Π_(n=1) ^∞ (1+(1/n^4 )) =?

$$\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\mathrm{1}+\frac{\mathrm{1}}{{n}^{\mathrm{4}} }\right)\:=?\: \\ $$

Question Number 142149 Answers: 5 Comments: 0

6. ∫(dx/(x^2 −3x+2)) 7. ∫((4dx)/(x^2 +2x+4)) 8. ∫((3−2xdx)/(x^2 −64)) 9. ∫((3x−1)/(x^3 +5x^2 +6x))dx 10. ∫((4−3x)/(x^3 −2x))dx 11. ∫(dx/(x^3 −2x+x))

$$\mathrm{6}.\:\int\frac{{dx}}{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}} \\ $$$$\mathrm{7}.\:\int\frac{\mathrm{4}{dx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{4}} \\ $$$$\mathrm{8}.\:\int\frac{\mathrm{3}−\mathrm{2}{xdx}}{{x}^{\mathrm{2}} −\mathrm{64}} \\ $$$$\mathrm{9}.\:\int\frac{\mathrm{3}{x}−\mathrm{1}}{{x}^{\mathrm{3}} +\mathrm{5}{x}^{\mathrm{2}} +\mathrm{6}{x}}{dx} \\ $$$$\mathrm{10}.\:\int\frac{\mathrm{4}−\mathrm{3}{x}}{{x}^{\mathrm{3}} −\mathrm{2}{x}}{dx} \\ $$$$\mathrm{11}.\:\int\frac{{dx}}{{x}^{\mathrm{3}} −\mathrm{2}{x}+{x}} \\ $$$$ \\ $$

Question Number 142148 Answers: 1 Comments: 0

Σ_(n=0) ^∞ (1/(n!))=?

$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!}=? \\ $$

Question Number 142140 Answers: 1 Comments: 0

[lim_(x→0) ((sin x)/x)]=? lim_(x→0) {[((100sin^(−1) x)/x)]+[((100tan^(−1) x)/x)]}=? lim_(x→0) {[((100x)/(sin^(−1) x))]+[((100x)/(tan^(−1) x))]}=? where [x] denotes greatest integer less than or equal to x. solution please

$$\left[\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{x}}{{x}}\right]=? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{\left[\frac{\mathrm{100sin}^{−\mathrm{1}} {x}}{{x}}\right]+\left[\frac{\mathrm{100tan}^{−\mathrm{1}} {x}}{{x}}\right]\right\}=? \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left\{\left[\frac{\mathrm{100}{x}}{\mathrm{sin}^{−\mathrm{1}} {x}}\right]+\left[\frac{\mathrm{100}{x}}{\mathrm{tan}^{−\mathrm{1}} {x}}\right]\right\}=? \\ $$$${where}\:\left[{x}\right]\:{denotes}\:{greatest}\:{integer}\: \\ $$$${less}\:{than}\:{or}\:{equal}\:{to}\:{x}. \\ $$$${solution}\:{please} \\ $$

Question Number 142138 Answers: 0 Comments: 6

(√(y^2 +1)) + (√(x^2 +4)) + (√(z^2 +9)) = 10 x+y+z=?

$$\sqrt{{y}^{\mathrm{2}} +\mathrm{1}}\:+\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}}\:+\:\sqrt{{z}^{\mathrm{2}} +\mathrm{9}}\:=\:\mathrm{10} \\ $$$${x}+{y}+{z}=? \\ $$

Question Number 142131 Answers: 1 Comments: 1

∫(dx/(3+2sinx+cosx))dx

$$\int\frac{{dx}}{\mathrm{3}+\mathrm{2}{sinx}+{cosx}}{dx} \\ $$

Question Number 142120 Answers: 2 Comments: 2