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IntegrationQuestion and Answers: Page 149

Question Number 110875 Answers: 4 Comments: 0

(1)∫_e ^e^e ((ln (x).ln (ln (x)))/x) dx ? (2)lim_(x→π/4) ((cosec^2 x−2)/(cot x−1)) (3) Given { ((xy=((16y−9x)/(45)))),(((4/( (√x)))−(3/( (√y))) = 5)) :} ⇒find 9(√(xy))

$$\left(\mathrm{1}\right)\underset{\mathrm{e}} {\overset{\mathrm{e}^{\mathrm{e}} } {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{x}\right).\mathrm{ln}\:\left(\mathrm{ln}\:\left(\mathrm{x}\right)\right)}{\mathrm{x}}\:\mathrm{dx}\:? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\:\frac{\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{2}}{\mathrm{cot}\:\mathrm{x}−\mathrm{1}} \\ $$$$\left(\mathrm{3}\right)\:\mathrm{Given}\:\begin{cases}{\mathrm{xy}=\frac{\mathrm{16y}−\mathrm{9x}}{\mathrm{45}}}\\{\frac{\mathrm{4}}{\:\sqrt{\mathrm{x}}}−\frac{\mathrm{3}}{\:\sqrt{\mathrm{y}}}\:=\:\mathrm{5}}\end{cases} \\ $$$$\Rightarrow\mathrm{find}\:\mathrm{9}\sqrt{\mathrm{xy}} \\ $$

Question Number 110800 Answers: 0 Comments: 0

∫((sin(x))/(x^2 +1))dx

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

Question Number 110772 Answers: 0 Comments: 0

lim_(n→∞) (1+Σ_(r=1) ^n (1/(3^r r!))Π_(k=1) ^r (2k−1))

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{3}^{{r}} {r}!}\underset{{k}=\mathrm{1}} {\overset{{r}} {\prod}}\left(\mathrm{2}{k}−\mathrm{1}\right)\right) \\ $$

Question Number 110749 Answers: 1 Comments: 0

please evaluate : Ω=∫_0 ^( (1/2)) ((ln^2 (1−x))/x) dx=??? M.N.July 1970# .... Good luck....

$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:{please}\:{evaluate}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}\:{dx}=???\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathscr{M}.\mathscr{N}.\mathscr{J}{uly}\:\mathrm{1970}# \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\mathscr{G}{ood}\:\:{luck}.... \\ $$$$ \\ $$$$ \\ $$

Question Number 118674 Answers: 1 Comments: 0

Please integrate ∫_0 ^1 (1/(1+x^c ))dx where c is a constant.

$${Please}\:{integrate} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{x}^{{c}} }{dx}\:{where}\:{c}\:{is}\:{a}\:{constant}. \\ $$

Question Number 110551 Answers: 2 Comments: 0

Question Number 110549 Answers: 2 Comments: 1

Question Number 110543 Answers: 1 Comments: 0

Question Number 110888 Answers: 3 Comments: 0

....calculus.... please solve : Ω_1 =∫_0 ^( (π/4)) ((√(tan(x))) +(√(cot(x))) )dx=?? Ω_2 =∫_0 ^(π/4) tan(x)ln((1+tan^2 (x)))dx =?? ...M.N.july 1970#... Good luck

$$\:\:\:\:\:\:\:\:\:\:....{calculus}.... \\ $$$${please}\:{solve}\:: \\ $$$$ \\ $$$$\Omega_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{4}}} \left(\sqrt{{tan}\left({x}\right)}\:+\sqrt{{cot}\left({x}\right)}\:\right){dx}=?? \\ $$$$\:\Omega_{\mathrm{2}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {tan}\left({x}\right){ln}\left(\left(\mathrm{1}+{tan}^{\mathrm{2}} \left({x}\right)\right)\right){dx}\:=?? \\ $$$$\:\: \\ $$$$\:\:\:\:\:\:\:\:...\mathscr{M}.\mathscr{N}.{july}\:\mathrm{1970}#... \\ $$$$\:\mathscr{G}{ood}\:{luck} \\ $$$$ \\ $$$$ \\ $$

Question Number 110451 Answers: 1 Comments: 0

calculate U_n =∫_([(1/n),n[^2 ) (x^2 −y^2 )e^(−x^2 −y^2 ) dxdy and lim_(n→+∞) U_n

$$\mathrm{calculate}\:\mathrm{U}_{\mathrm{n}} =\int_{\left[\frac{\mathrm{1}}{\mathrm{n}},\mathrm{n}\left[^{\mathrm{2}} \right.\right.} \:\:\:\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} \right)\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} } \mathrm{dxdy} \\ $$$$\mathrm{and}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{U}_{\mathrm{n}} \\ $$

Question Number 110450 Answers: 1 Comments: 0

find ∫∫_([0,1]^2 ) ln(x^2 +3y^2 ) e^(−x^2 −3y^2 ) dxdy

$$\mathrm{find}\:\int\int_{\left[\mathrm{0},\mathrm{1}\right]^{\mathrm{2}} } \:\:\:\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{3y}^{\mathrm{2}} \right)\:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} −\mathrm{3y}^{\mathrm{2}} } \:\mathrm{dxdy} \\ $$

Question Number 110448 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) ((cos(2x))/((x^2 −4i)^3 ))dx (i=(√(−1)))

$$\mathrm{calculate}\:\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{4i}\right)^{\mathrm{3}} }\mathrm{dx}\:\:\:\:\:\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 110447 Answers: 1 Comments: 0

calculate ∫_(−∞) ^(+∞) (dx/((x^2 −ix +1)^2 )) (i=(√(−1)))

$$\mathrm{calculate}\:\int_{−\infty} ^{+\infty} \:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} −\mathrm{ix}\:+\mathrm{1}\right)^{\mathrm{2}} }\:\:\left(\mathrm{i}=\sqrt{−\mathrm{1}}\right) \\ $$

Question Number 110301 Answers: 0 Comments: 1

Question Number 110262 Answers: 0 Comments: 0

Solve for X(x,y,z), Y(x,y,z), Z(x,y,z) { (((∂Z/∂y)−(∂Y/∂z)=1−x^2 )),(((∂Z/∂x)−(∂X/∂z)=−(y^2 /2))),(((∂Y/∂x)−(∂X/∂y)=z(2x−y))) :} where { ((X(x,y,0)=0)),((Y(x,y,0)=0)),((Z(x,y,0)=0)) :}

$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right),\:\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right) \\ $$$$\begin{cases}{\frac{\partial\mathrm{Z}}{\partial\mathrm{y}}−\frac{\partial\mathrm{Y}}{\partial\mathrm{z}}=\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\\{\frac{\partial\mathrm{Z}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{z}}=−\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{2}}}\\{\frac{\partial\mathrm{Y}}{\partial\mathrm{x}}−\frac{\partial\mathrm{X}}{\partial\mathrm{y}}=\mathrm{z}\left(\mathrm{2x}−\mathrm{y}\right)}\end{cases}\:\mathrm{where}\:\begin{cases}{\mathrm{X}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Y}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\\{\mathrm{Z}\left(\mathrm{x},\mathrm{y},\mathrm{0}\right)=\mathrm{0}}\end{cases} \\ $$

Question Number 110247 Answers: 1 Comments: 0

Let f(x) = ∫_0 ^( x) e^(−t) dt then f ′′(x) = ??

$$\mathrm{Let}\:{f}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{\:{x}} {e}^{−{t}} {dt}\: \\ $$$$\mathrm{then}\:{f}\:''\left({x}\right)\:=\:?? \\ $$

Question Number 110245 Answers: 1 Comments: 2

solve ∫(dx/( ((c−(√(b−ax))))^(1/3) ))

$${solve}\:\int\frac{{dx}}{\:\sqrt[{\mathrm{3}}]{{c}−\sqrt{{b}−{ax}}}} \\ $$

Question Number 110233 Answers: 0 Comments: 0