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IntegrationQuestion and Answers: Page 114

Question Number 129150 Answers: 1 Comments: 0

∫_0 ^( x) (x−u)^2 u f(u) du = 4x−6sin x+2xcos x f(x)=?

$$\:\int_{\mathrm{0}} ^{\:\mathrm{x}} \left(\mathrm{x}−\mathrm{u}\right)^{\mathrm{2}} \mathrm{u}\:\mathrm{f}\left(\mathrm{u}\right)\:\mathrm{du}\:=\:\mathrm{4x}−\mathrm{6sin}\:\mathrm{x}+\mathrm{2xcos}\:\mathrm{x} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=? \\ $$

Question Number 129068 Answers: 1 Comments: 0

Question Number 129064 Answers: 1 Comments: 0

valuate the following integral I=∫_1 ^∞ (dt/((t)^(2k+v+(3/2)) (√(t−1)))) and prove that: I=(√π)((Γ(v+1))/(Γ(v+(3/2)))) ((((((v+1)/2))_k (1+(v/2))_k )/(((v/2)+(3/4))_k ((v/2)+(5/4))_k )))

$${valuate}\:{the}\:{following}\:{integral} \\ $$$${I}=\int_{\mathrm{1}} ^{\infty} \frac{{dt}}{\left({t}\right)^{\mathrm{2}{k}+{v}+\frac{\mathrm{3}}{\mathrm{2}}} \sqrt{{t}−\mathrm{1}}} \\ $$$${and}\:{prove}\:{that}: \\ $$$${I}=\sqrt{\pi}\frac{\Gamma\left({v}+\mathrm{1}\right)}{\Gamma\left({v}+\frac{\mathrm{3}}{\mathrm{2}}\right)}\:\left(\frac{\left(\frac{{v}+\mathrm{1}}{\mathrm{2}}\right)_{{k}} \left(\mathrm{1}+\frac{{v}}{\mathrm{2}}\right)_{{k}} }{\left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{4}}\right)_{{k}} \left(\frac{{v}}{\mathrm{2}}+\frac{\mathrm{5}}{\mathrm{4}}\right)_{{k}} }\right) \\ $$

Question Number 129060 Answers: 1 Comments: 0

φ = ∫ ((ln (x))/x^2 ) dx

$$\:\phi\:=\:\int\:\frac{\mathrm{ln}\:\left(\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} }\:\mathrm{dx}\: \\ $$

Question Number 129057 Answers: 2 Comments: 0

∫_1 ^∞ (1/(1+x^3 ))dx = ...

$$\: \\ $$$$\:\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{1}}{\mathrm{1}+\mathrm{x}^{\mathrm{3}} }\mathrm{dx}\:=\:... \\ $$

Question Number 129038 Answers: 1 Comments: 0

∫ ((5e^(4t) +10e^(2t) +2)/(e^(2t) +2)) dt

$$\int\:\frac{\mathrm{5e}^{\mathrm{4t}} +\mathrm{10e}^{\mathrm{2t}} +\mathrm{2}}{\mathrm{e}^{\mathrm{2t}} +\mathrm{2}}\:\mathrm{dt}\: \\ $$

Question Number 129033 Answers: 0 Comments: 1

prove using the first principle that the derivative of sin x is cox x and that the derivative of cos x is −sinx

$${prove}\:{using}\:{the}\:{first}\:{principle}\:{that} \\ $$$${the}\:{derivative}\:{of}\:\boldsymbol{{sin}}\:\boldsymbol{{x}}\:{is}\:\boldsymbol{{cox}}\:\boldsymbol{{x}}\:{and} \\ $$$${that}\:{the}\:{derivative}\:{of}\:\boldsymbol{{cos}}\:\boldsymbol{{x}}\:{is} \\ $$$$−\boldsymbol{{sinx}} \\ $$

Question Number 129031 Answers: 1 Comments: 0

Given that tan^(−1) x show that (dy/dx) = (1/(1+x^2 ))

$${Given}\:{that}\:\boldsymbol{{tan}}^{−\mathrm{1}} \boldsymbol{{x}}\:{show}\:{that}\:\: \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}\:=\:\frac{\mathrm{1}}{\mathrm{1}+\boldsymbol{{x}}^{\mathrm{2}} } \\ $$

Question Number 129014 Answers: 2 Comments: 1

∫_(−(1/2)) ^( −(1/4)) x(x+1) (√(1+(1/x^2 )+(1/((x+1)^2 )))) dx =?

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

Question Number 129009 Answers: 1 Comments: 0

Given f(x+1)=((1+f(x))/(1−f(x))) ; f(2)=2 and ∫_2 ^( 2018) x.f(2018)dx=2^a .3^b .5^c .7^d .11^e .101^f then a+b+c+d+e +f=__

$$\:\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}+\mathrm{1}\right)=\frac{\mathrm{1}+\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{1}−\mathrm{f}\left(\mathrm{x}\right)}\:;\:\mathrm{f}\left(\mathrm{2}\right)=\mathrm{2} \\ $$$$\mathrm{and}\:\int_{\mathrm{2}} ^{\:\mathrm{2018}} \mathrm{x}.\mathrm{f}\left(\mathrm{2018}\right)\mathrm{dx}=\mathrm{2}^{\mathrm{a}} .\mathrm{3}^{\mathrm{b}} .\mathrm{5}^{\mathrm{c}} .\mathrm{7}^{\mathrm{d}} .\mathrm{11}^{\mathrm{e}} .\mathrm{101}^{\mathrm{f}} \\ $$$$\mathrm{then}\:\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}\:+\mathrm{f}=\_\_ \\ $$

Question Number 129001 Answers: 2 Comments: 0

∫_1 ^∞ (dx/(1+x^4 )) = ...

$$\: \\ $$$$\:\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{dx}}{\mathrm{1}+\mathrm{x}^{\mathrm{4}} \:}\:=\:... \\ $$

Question Number 128985 Answers: 0 Comments: 0

∫ ln (tan x) ln (sin x) dx ?

$$\:\int\:\mathrm{ln}\:\left(\mathrm{tan}\:\mathrm{x}\right)\:\mathrm{ln}\:\left(\mathrm{sin}\:\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 128983 Answers: 1 Comments: 1

∫ ((√x)/(x−1)) dx =?

$$\:\int\:\frac{\sqrt{\mathrm{x}}}{\mathrm{x}−\mathrm{1}}\:\mathrm{dx}\:=? \\ $$

Question Number 128982 Answers: 1 Comments: 0

∫_( 0) ^( ∞) e^(−x^2 ) cos x dx

$$\:\int_{\:\mathrm{0}} ^{\:\infty} \:\mathrm{e}^{−\mathrm{x}^{\mathrm{2}} } \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx} \\ $$

Question Number 128977 Answers: 3 Comments: 0

... nice calculus... Evaluate ::: I:=∫_0 ^( 1) ln((1/x) − x)(dx/(1+x^2 )) =? ∗....................................∗

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{nice}\:\:{calculus}... \\ $$$$\:\:\:\:\mathscr{E}{valuate}\:::: \\ $$$$\:\:\:\:\:\:\mathrm{I}:=\int_{\mathrm{0}} ^{\:\mathrm{1}} {ln}\left(\frac{\mathrm{1}}{{x}}\:−\:{x}\right)\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} }\:=? \\ $$$$\:\:\:\:\:\:\:\ast....................................\ast \\ $$

Question Number 128956 Answers: 3 Comments: 0

calculate ∫_0 ^π ((sin(2x))/(3+cos(4x)))dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\pi} \:\frac{\mathrm{sin}\left(\mathrm{2x}\right)}{\mathrm{3}+\mathrm{cos}\left(\mathrm{4x}\right)}\mathrm{dx} \\ $$

Question Number 128955 Answers: 3 Comments: 0

find ∫_0 ^∞ e^(−3x) ln(1+e^(2x) )dx

$$\mathrm{find}\:\int_{\mathrm{0}} ^{\infty} \:\mathrm{e}^{−\mathrm{3x}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{2x}} \right)\mathrm{dx} \\ $$

Question Number 128954 Answers: 1 Comments: 0

calculate f(λ) =∫_0 ^1 ln(x^4 +λ^4 )dx with λ>0 then find the value of ∫_0 ^1 ln(1+x^4 )dx

$$\mathrm{calculate}\:\:\mathrm{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{x}^{\mathrm{4}} \:+\lambda^{\mathrm{4}} \right)\mathrm{dx}\:\:\:\mathrm{with}\:\lambda>\mathrm{0}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{4}} \right)\mathrm{dx} \\ $$

Question Number 128953 Answers: 1 Comments: 0

calculate ∫_0 ^1 ln(1+x^6 )dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{ln}\left(\mathrm{1}+\mathrm{x}^{\mathrm{6}} \right)\mathrm{dx} \\ $$

Question Number 128952 Answers: 3 Comments: 0

calculate A_n =∫_0 ^∞ (dx/((x^2 +1)^n )) , n integr and n≥1

$$\mathrm{calculate}\:\mathrm{A}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{n}} }\:\:\:,\:\mathrm{n}\:\mathrm{integr}\:\mathrm{and}\:\mathrm{n}\geqslant\mathrm{1} \\ $$

Question Number 128950 Answers: 0 Comments: 0

calculate f(a) =∫_0 ^∞ (dx/((x^2 +a)^2 (x^2 −x+1)^2 )) with a>0

$$\mathrm{calculate}\:\mathrm{f}\left(\mathrm{a}\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{dx}}{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{a}\right)^{\mathrm{2}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} }\:\:\mathrm{with}\:\mathrm{a}>\mathrm{0} \\ $$

Question Number 128943 Answers: 0 Comments: 0

∫ (x^9 /((x^2 +m^2 )sec (5x))) dx ?

$$\:\int\:\frac{\mathrm{x}^{\mathrm{9}} }{\left(\mathrm{x}^{\mathrm{2}} +\mathrm{m}^{\mathrm{2}} \right)\mathrm{sec}\:\left(\mathrm{5x}\right)}\:\mathrm{dx}\:? \\ $$

Question Number 128926 Answers: 0 Comments: 0

β = ∫_0 ^( 5) (dx/(x^3 (1+e^x )^2 )) ?

$$\:\beta\:=\:\int_{\mathrm{0}} ^{\:\mathrm{5}} \:\frac{{dx}}{{x}^{\mathrm{3}} \left(\mathrm{1}+{e}^{{x}} \right)^{\mathrm{2}} }\:? \\ $$

Question Number 128922 Answers: 1 Comments: 0

...advanced calculus... evaluate ::: Ω=∫_(0 ) ^( (1/2)) ((Arctanh(x))/x)dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:{calculus}... \\ $$$$\:\:\:\:\:{evaluate}\::::\: \\ $$$$\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}\:} ^{\:\frac{\mathrm{1}}{\mathrm{2}}} \frac{\mathrm{Arctanh}\left({x}\right)}{{x}}{dx}=? \\ $$$$ \\ $$

Question Number 128900 Answers: 3 Comments: 0

∫ ((4x+5)/((x+2)(x+3)(x+4)(x+5)+1)) dx?

$$\:\int\:\frac{\mathrm{4x}+\mathrm{5}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)\left(\mathrm{x}+\mathrm{4}\right)\left(\mathrm{x}+\mathrm{5}\right)+\mathrm{1}}\:\mathrm{dx}?\: \\ $$

Question Number 128888 Answers: 1 Comments: 0

∫ (((x−1)(x−2)(x−3))/((x−4)(x−5)(x−6))) dx =?

$$\:\int\:\frac{\left(\mathrm{x}−\mathrm{1}\right)\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)}{\left(\mathrm{x}−\mathrm{4}\right)\left(\mathrm{x}−\mathrm{5}\right)\left(\mathrm{x}−\mathrm{6}\right)}\:\mathrm{dx}\:=? \\ $$

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