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IntegrationQuestion and Answers: Page 79

Question Number 145646 Answers: 1 Comments: 0

Find the arc lenght of the function y^2 = (x^3 /a) where a is a constant for 0≤x≤((7a)/3)

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{arc}\:\mathrm{lenght}\:\mathrm{of}\:\mathrm{the}\:\mathrm{function}\:{y}^{\mathrm{2}} \:=\:\frac{{x}^{\mathrm{3}} }{{a}}\:\mathrm{where}\:{a}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{for} \\ $$$$\mathrm{0}\leqslant{x}\leqslant\frac{\mathrm{7}{a}}{\mathrm{3}} \\ $$

Question Number 145645 Answers: 0 Comments: 0

∫_0 ^a x^(−(x/a)) dx

$$\int_{\mathrm{0}} ^{{a}} {x}^{−\frac{{x}}{{a}}} {dx} \\ $$

Question Number 145588 Answers: 3 Comments: 0

Question Number 146212 Answers: 1 Comments: 0

K=∫(1/( (√(1+x^3 ))))dx

$$\mathrm{K}=\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }}\mathrm{dx} \\ $$

Question Number 145636 Answers: 2 Comments: 0

calculate ∫_0 ^∞ ((arctanx)/((1+x^2 )^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctanx}}{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 145635 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((arctan(3x^2 ))/(1+x^2 ))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{arctan}\left(\mathrm{3x}^{\mathrm{2}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx} \\ $$

Question Number 145549 Answers: 1 Comments: 0

Find the equation of the asymptotes to the curve y = f(x) where f(x) = ln(((x+3)/(x−1))) .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{asymptotes}\:\mathrm{to}\:\mathrm{the}\:\mathrm{curve} \\ $$$$\:{y}\:=\:{f}\left({x}\right)\:\mathrm{where}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\frac{{x}+\mathrm{3}}{{x}−\mathrm{1}}\right)\:.\: \\ $$

Question Number 145547 Answers: 1 Comments: 0

I_(m,n) = ∫_0 ^1 (1−x^m )^n dx Show that I_(m,n) (mn+1) = I_(m,n−1)

$${I}_{{m},{n}} \:=\:\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{{m}} \right)^{{n}} {dx} \\ $$$$\mathrm{Show}\:\mathrm{that}\:{I}_{{m},{n}} \left({mn}+\mathrm{1}\right)\:=\:{I}_{{m},{n}−\mathrm{1}} \\ $$

Question Number 145519 Answers: 0 Comments: 1

Question Number 145517 Answers: 1 Comments: 0

Find the center of mass for the thin plate bounded by curves g(x)=(x/2) and f(x)=(√x) , 0≤x≤1 .

$${Find}\:{the}\:{center}\:{of}\:{mass}\:{for}\: \\ $$$${the}\:{thin}\:{plate}\:{bounded}\:{by}\: \\ $$$${curves}\:{g}\left({x}\right)=\frac{{x}}{\mathrm{2}}\:{and}\:{f}\left({x}\right)=\sqrt{{x}} \\ $$$$,\:\mathrm{0}\leqslant{x}\leqslant\mathrm{1}\:. \\ $$

Question Number 145456 Answers: 1 Comments: 0

∫sin(x^2 +2)dx

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

Question Number 145385 Answers: 1 Comments: 0

∫_0 ^( (π/2)) ((1+cos (2x))/(sin (2x ))). ln((sec (x)))^(1/3) dx=?

$$\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \frac{\mathrm{1}+\mathrm{cos}\:\left(\mathrm{2x}\right)}{\mathrm{sin}\:\left(\mathrm{2x}\:\right)}.\:\mathrm{ln}\sqrt[{\mathrm{3}}]{\mathrm{sec}\:\left(\mathrm{x}\right)}\:\mathrm{dx}=? \\ $$

Question Number 145378 Answers: 1 Comments: 0

∫_0 ^( ∞) ((√( 1+ x^4 )) −x^( 2) )dx=?

$$ \\ $$$$\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\left(\sqrt{\:\mathrm{1}+\:\mathrm{x}^{\mathrm{4}} }\:−\mathrm{x}^{\:\mathrm{2}} \:\right)\mathrm{dx}=? \\ $$

Question Number 145361 Answers: 0 Comments: 0

∫_0 ^(+∞) ((t^2 +3t+3)/((t+1)^3 )) e^(−t) cos(t) dt

$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{t}^{\mathrm{2}} +\mathrm{3t}+\mathrm{3}}{\left(\mathrm{t}+\mathrm{1}\right)^{\mathrm{3}} }\:\mathrm{e}^{−\mathrm{t}} \mathrm{cos}\left(\mathrm{t}\right)\:\mathrm{dt} \\ $$

Question Number 145358 Answers: 1 Comments: 0

Evaluate:: ∫_0 ^1 ln(1+x^2 )∙arctan(x)dx=?

$$\mathrm{Evaluate}::\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)\centerdot\mathrm{arctan}\left(\mathrm{x}\right)\mathrm{dx}=? \\ $$

Question Number 145345 Answers: 1 Comments: 0

Σ_(n=1) ^∞ (((−1)^n n)/((2n+1)!))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{\mathrm{n}} \mathrm{n}}{\left(\mathrm{2n}+\mathrm{1}\right)!}=? \\ $$

Question Number 145339 Answers: 1 Comments: 0

Let f(x)=e^x cos x,Find Σ_(n=0) ^∞ ((f^((n)) (x))/2^n )=?

$$\mathrm{Let}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{e}^{\mathrm{x}} \mathrm{cos}\:\mathrm{x},\mathrm{Find}\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{x}\right)}{\mathrm{2}^{\mathrm{n}} }=? \\ $$

Question Number 145319 Answers: 1 Comments: 0

# Calculus# Σ_(n=0) ^∞ (1/(n! + (n + 1 )!)) =?

$$\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:#\:\:\mathrm{Calculus}# \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{{n}!\:+\:\left({n}\:+\:\mathrm{1}\:\right)!}\:=? \\ $$$$ \\ $$

Question Number 145274 Answers: 1 Comments: 0

∫_(∣z∣=1) ((f^− (z))/(z−a))dz

$$\int_{\mid{z}\mid=\mathrm{1}} \frac{\overset{−} {{f}}\left({z}\right)}{{z}−{a}}{dz} \\ $$

Question Number 145270 Answers: 1 Comments: 0

# Calculus ( I ) # Σ_(n=1) ^∞ Arccot(3 +((n ( n + 1))/3) )= ? .....

$$ \\ $$$$\:\:\:\:\:\:\:#\:\mathrm{Calculus}\:\left(\:\mathrm{I}\:\right)\:# \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\mathrm{Arccot}\left(\mathrm{3}\:+\frac{{n}\:\left(\:{n}\:+\:\mathrm{1}\right)}{\mathrm{3}}\:\right)=\:? \\ $$$$\:\:\:\:\:\:..... \\ $$

Question Number 145202 Answers: 2 Comments: 1

Question Number 145200 Answers: 1 Comments: 0

evaluate:: Σ_(n=0) ^∞ (1/(n!(n^4 +n^2 +1)))=(e/2)

$$\mathrm{evaluate}::\:\:\underset{\mathrm{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{n}!\left(\mathrm{n}^{\mathrm{4}} +\mathrm{n}^{\mathrm{2}} +\mathrm{1}\right)}=\frac{\mathrm{e}}{\mathrm{2}} \\ $$

Question Number 145184 Answers: 0 Comments: 0

find lim_(x→0) ((sin(sh(2x))−sh(sin(3x)))/x^2 )

$$\mathrm{find}\:\mathrm{lim}_{\mathrm{x}\rightarrow\mathrm{0}} \:\:\frac{\mathrm{sin}\left(\mathrm{sh}\left(\mathrm{2x}\right)\right)−\mathrm{sh}\left(\mathrm{sin}\left(\mathrm{3x}\right)\right)}{\mathrm{x}^{\mathrm{2}} } \\ $$

Question Number 145183 Answers: 2 Comments: 1

find ∫_0 ^1 (dx/(((√x)+(√(x+1)))^3 ))

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{\mathrm{dx}}{\left(\sqrt{\mathrm{x}}+\sqrt{\mathrm{x}+\mathrm{1}}\right)^{\mathrm{3}} } \\ $$

Question Number 145081 Answers: 1 Comments: 0

∫_0 ^∞ (x−(x^3 /2)+(x^5 /(2∙4))−(x^7 /(2∙4∙6))+...)∙(1+(x^2 /2^2 )+(x^4 /(2^2 ∙4^2 ))+(x^6 /(2^2 ∙4^2 ∙6^2 ))+...)dx=(√e)

$$\int_{\mathrm{0}} ^{\infty} \left(\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{2}}+\frac{\mathrm{x}^{\mathrm{5}} }{\mathrm{2}\centerdot\mathrm{4}}−\frac{\mathrm{x}^{\mathrm{7}} }{\mathrm{2}\centerdot\mathrm{4}\centerdot\mathrm{6}}+...\right)\centerdot\left(\mathrm{1}+\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}^{\mathrm{2}} }+\frac{\mathrm{x}^{\mathrm{4}} }{\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{4}^{\mathrm{2}} }+\frac{\mathrm{x}^{\mathrm{6}} }{\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{4}^{\mathrm{2}} \centerdot\mathrm{6}^{\mathrm{2}} }+...\right)\mathrm{dx}=\sqrt{\mathrm{e}} \\ $$

Question Number 145064 Answers: 1 Comments: 0

∫cos 2xln (1+tan x)dx

$$\:\:\:\:\:\:\int\mathrm{cos}\:\mathrm{2xln}\:\left(\mathrm{1}+\mathrm{tan}\:\mathrm{x}\right)\mathrm{dx} \\ $$$$\:\:\:\:\:\: \\ $$

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