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IntegrationQuestion and Answers: Page 150

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

Question Number 110218 Answers: 1 Comments: 0

Question Number 110215 Answers: 1 Comments: 1

Question Number 110203 Answers: 1 Comments: 0

Question Number 110143 Answers: 2 Comments: 0

Question Number 110027 Answers: 2 Comments: 0

Question Number 109989 Answers: 1 Comments: 1

Question Number 109988 Answers: 0 Comments: 0

Question Number 109949 Answers: 1 Comments: 0

Question Number 109891 Answers: 1 Comments: 0

((Δbe▽)/(math)) ∫ ((arc tan x)/((1+(1/x^2 )))) dx ?

$$\:\:\frac{\Delta{be}\bigtriangledown}{{math}} \\ $$$$\int\:\frac{\mathrm{arc}\:\mathrm{tan}\:{x}}{\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)}\:{dx}\:? \\ $$

Question Number 109884 Answers: 1 Comments: 1

Question Number 109872 Answers: 2 Comments: 1

((★be★)/(Math)) ∫ ((cos x dx)/(sin^2 x+4sin x−5)) ?

$$\:\:\frac{\bigstar{be}\bigstar}{\mathcal{M}{ath}} \\ $$$$\int\:\frac{\mathrm{cos}\:{x}\:{dx}}{\mathrm{sin}\:^{\mathrm{2}} {x}+\mathrm{4sin}\:{x}−\mathrm{5}}\:? \\ $$

Question Number 109849 Answers: 0 Comments: 0

Question Number 109839 Answers: 4 Comments: 1

((JS)/(≈♥≈)) (1) ∫ (((√(x+1))−(√(x−1)))/( (√(x+1))+(√(x−1)))) dx (2) ∫ ((√(tan x))/(1+(√(tan x)) )) dx

$$\:\frac{{JS}}{\approx\heartsuit\approx} \\ $$$$\left(\mathrm{1}\right)\:\int\:\frac{\sqrt{{x}+\mathrm{1}}−\sqrt{{x}−\mathrm{1}}}{\:\sqrt{{x}+\mathrm{1}}+\sqrt{{x}−\mathrm{1}}}\:{dx} \\ $$$$\left(\mathrm{2}\right)\:\int\:\frac{\sqrt{\mathrm{tan}\:{x}}}{\mathrm{1}+\sqrt{\mathrm{tan}\:{x}}\:}\:{dx}\: \\ $$

Question Number 109838 Answers: 1 Comments: 0

please prove::: ∫_0 ^( 1) (1/( (√(1−x))))log((x/(1−x)))dx =4log(2)

$$ \\ $$$$ \\ $$$$\:\:\:\:{please}\:{prove}::: \\ $$$$\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}}}{log}\left(\frac{{x}}{\mathrm{1}−{x}}\right){dx}\:=\mathrm{4}{log}\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$

Question Number 109801 Answers: 0 Comments: 2

calculste ∫_0 ^1 (√(1+x^6 ))dx

$$\mathrm{calculste}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{6}} }\mathrm{dx} \\ $$

Question Number 109794 Answers: 0 Comments: 0

Question Number 109709 Answers: 2 Comments: 0

∫_(0 ) ^(π/4) ln(tanx+1)dx

$$\:\:\:\:\:\int_{\mathrm{0}\:} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{tanx}+\mathrm{1}\right)\mathrm{dx} \\ $$

Question Number 109658 Answers: 1 Comments: 0

Question Number 109848 Answers: 0 Comments: 0

Question Number 109642 Answers: 1 Comments: 0

Question Number 109616 Answers: 1 Comments: 1

find ∫_0 ^1 (√(1+x^4 ))dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{4}} }\mathrm{dx} \\ $$

Question Number 109546 Answers: 1 Comments: 2

If f(x) continue in [ 1,30] and ∫_6 ^(30) f(x)dx = 30, then ∫_1 ^9 f(3y+3)dy = __

$${If}\:{f}\left({x}\right)\:{continue}\:{in}\:\left[\:\mathrm{1},\mathrm{30}\right]\:{and}\: \\ $$$$\underset{\mathrm{6}} {\overset{\mathrm{30}} {\int}}{f}\left({x}\right){dx}\:=\:\mathrm{30},\:{then}\:\underset{\mathrm{1}} {\overset{\mathrm{9}} {\int}}{f}\left(\mathrm{3}{y}+\mathrm{3}\right){dy}\:=\:\_\_ \\ $$

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