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IntegrationQuestion and Answers: Page 79

Question Number 146110 Answers: 2 Comments: 0

∫((x+1)/(2x^2 +x+1))dx

$$\int\frac{\mathrm{x}+\mathrm{1}}{\mathrm{2x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 146108 Answers: 0 Comments: 0

Solve in Z[X] 1) XP ′ ≡ −1 mod(X^4 +1) 2) X^3 P −P ′ ≡ 1−X^2 mod(X^4 +1) 3) P^2 −X^3 P−X^2 ≡ 0 mod(X^2 +2)

$$\:{Solve}\:\:{in}\:\mathbb{Z}\left[{X}\right] \\ $$$$\left.\mathrm{1}\right)\:{XP}\:'\:\equiv\:−\mathrm{1}\:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{2}\right)\:{X}^{\mathrm{3}} {P}\:−{P}\:'\:\equiv\:\mathrm{1}−{X}^{\mathrm{2}} \:{mod}\left({X}^{\mathrm{4}} +\mathrm{1}\right) \\ $$$$\left.\mathrm{3}\right)\:{P}\:^{\mathrm{2}} −{X}^{\mathrm{3}} {P}−{X}^{\mathrm{2}} \:\:\equiv\:\mathrm{0}\:{mod}\left({X}^{\mathrm{2}} +\mathrm{2}\right) \\ $$

Question Number 146131 Answers: 1 Comments: 1

Question Number 146090 Answers: 2 Comments: 0

∫_0 ^∞ ((sinh(at)sinh(bt))/(sinh(ct)e^(tz) ))dt= ((ab)/(c(z^2 +c^2 −a^2 −b^2 +K_(k=1) ^∞ ((−4k^2 (k^2 c^2 −a^2 )(k^2 c^2 −b^2 ))/((2k+1)(z^2 +(2k^2 +2k+1)c^2 −a^2 −b^2 ))))))

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{sinh}\left(\mathrm{at}\right)\mathrm{sinh}\left(\mathrm{bt}\right)}{\mathrm{sinh}\left(\mathrm{ct}\right)\mathrm{e}^{\mathrm{tz}} }\mathrm{dt}= \\ $$$$\frac{\mathrm{ab}}{\mathrm{c}\left(\mathrm{z}^{\mathrm{2}} +\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} +\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\mathrm{K}}}\frac{−\mathrm{4k}^{\mathrm{2}} \left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} \right)\left(\mathrm{k}^{\mathrm{2}} \mathrm{c}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}{\left(\mathrm{2k}+\mathrm{1}\right)\left(\mathrm{z}^{\mathrm{2}} +\left(\mathrm{2k}^{\mathrm{2}} +\mathrm{2k}+\mathrm{1}\right)\mathrm{c}^{\mathrm{2}} −\mathrm{a}^{\mathrm{2}} −\mathrm{b}^{\mathrm{2}} \right)}\right)} \\ $$

Question Number 146073 Answers: 0 Comments: 0

Let K be nonempty corps , K^∗ =K−{0_K } Prove that 1) Π_(x∈K^∗ ) x = −1 2)Deduce that p is prime ⇔ (p−1)!≡−1[p]

$${Let}\:{K}\:{be}\:{nonempty}\:\:{corps}\:,\:{K}^{\ast} ={K}−\left\{\mathrm{0}_{{K}} \right\} \\ $$$${Prove}\:{that} \\ $$$$\left.\mathrm{1}\right)\:\underset{{x}\in{K}^{\ast} } {\prod}{x}\:=\:−\mathrm{1} \\ $$$$\left.\mathrm{2}\right){Deduce}\:{that}\: \\ $$$$\:\:{p}\:{is}\:{prime}\:\Leftrightarrow\:\left({p}−\mathrm{1}\right)!\equiv−\mathrm{1}\left[{p}\right] \\ $$

Question Number 146067 Answers: 1 Comments: 0

transform the cartesian inyegral ∫_0 ^1 ∫_0 ^(√(1−x^2 )) e^(−(x^2 +y^2 )) dy dx into polar integral and evaluate it.

$${transform}\:{the}\:{cartesian}\:{inyegral}\: \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\:\:\underset{\mathrm{0}} {\overset{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }} {\int}}{e}^{−\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right)} \:{dy}\:{dx}\:{into}\:{polar}\:{integral}\: \\ $$$${and}\:{evaluate}\:{it}. \\ $$

Question Number 146062 Answers: 0 Comments: 1

find values a , b , c such that: −1≤ ax^2 +bx +c ≤ 1 and ((6b^( 2) + 8 a^( 2) )/3) is Max...

$$ \\ $$$$\:\:\:\:{find}\:\:{values}\:\:{a}\:,\:{b}\:,\:{c}\:\:{such}\:{that}: \\ $$$$\:\:\:\:−\mathrm{1}\leqslant\:{ax}\:^{\mathrm{2}} +{bx}\:+{c}\:\leqslant\:\mathrm{1} \\ $$$$\:\:\:\:\:\:{and}\:\:\frac{\mathrm{6}{b}^{\:\mathrm{2}} +\:\mathrm{8}\:{a}^{\:\mathrm{2}} }{\mathrm{3}}\:{is}\:{Max}... \\ $$

Question Number 146035 Answers: 2 Comments: 0

help me please ∫((ln(x+1))/x)dx=??

$${help}\:{me}\:{please} \\ $$$$\int\frac{{ln}\left({x}+\mathrm{1}\right)}{{x}}{dx}=?? \\ $$$$ \\ $$

Question Number 145888 Answers: 1 Comments: 0

Question Number 145828 Answers: 0 Comments: 0

∫(1/(x^α +a))dx

$$\int\frac{\mathrm{1}}{{x}^{\alpha} +{a}}{dx} \\ $$

Question Number 145827 Answers: 2 Comments: 0

Use Abel summation to evaluate :: Σ_(n=1) ^∞ (1/((2n−1)∙2^n ))=(1/( (√2)))ln((√2)+1)

$$\mathrm{Use}\:\mathrm{Abel}\:\mathrm{summation}\:\mathrm{to}\:\mathrm{evaluate}\::: \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left(\mathrm{2n}−\mathrm{1}\right)\centerdot\mathrm{2}^{\mathrm{n}} }=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\mathrm{ln}\left(\sqrt{\mathrm{2}}+\mathrm{1}\right) \\ $$

Question Number 145779 Answers: 1 Comments: 0

lim_(x→0) ∫_0 ^1 (e^t +e^(−t) −2)(dt/(1−cosx))

$${li}\underset{{x}\rightarrow\mathrm{0}} {{m}}\int_{\mathrm{0}} ^{\mathrm{1}} \left({e}^{{t}} +{e}^{−{t}} −\mathrm{2}\right)\frac{{dt}}{\mathrm{1}−{cosx}} \\ $$

Question Number 145777 Answers: 2 Comments: 0

∫(1/( (√(1−9x^2 ))))dx

$$\int\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−\mathrm{9}{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 145776 Answers: 2 Comments: 0

∫_0 ^(π/2) e^x cosxdx

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {e}^{{x}} {cosxdx} \\ $$

Question Number 145775 Answers: 2 Comments: 0

∫((2x+1)/( (√(x^2 +4x+5))))dx

$$\int\frac{\mathrm{2}{x}+\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{5}}}{dx} \\ $$

Question Number 145745 Answers: 1 Comments: 0

calculate ∫_0 ^∞ ((cos(2x))/((x^2 +1)^2 (x^2 +4)))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{2x}\right)}{\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{4}\right)}\mathrm{dx} \\ $$

Question Number 145723 Answers: 1 Comments: 0

Question Number 145722 Answers: 0 Comments: 0

Question Number 145676 Answers: 2 Comments: 0

... advanced ......calculus... prove that: Σ_(n=1) ^∞ (((−1)^(n−1) )/(n^( 3) ((( 2n)),(( n)) ))) = (2/5) ζ (3 )

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{advanced}\:......{calculus}... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:{prove}\:{that}:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} }{{n}^{\:\mathrm{3}} \:\begin{pmatrix}{\:\mathrm{2}{n}}\\{\:\:\:{n}}\end{pmatrix}}\:=\:\frac{\mathrm{2}}{\mathrm{5}}\:\zeta\:\left(\mathrm{3}\:\right) \\ $$$$ \\ $$

Question Number 145671 Answers: 6 Comments: 1

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

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

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