Question and Answers Forum

All Questions Topic List

IntegrationQuestion and Answers: Page 107

Question Number 132121 Answers: 2 Comments: 2

∫_2 ^8 ((√x)/( (√(10−x)) +(√x))) dx

$$\int_{\mathrm{2}} ^{\mathrm{8}} \frac{\sqrt{{x}}}{\:\sqrt{\mathrm{10}−{x}}\:+\sqrt{{x}}}\:\mathrm{dx} \\ $$

Question Number 132110 Answers: 1 Comments: 0

∫(1/(ln x))dx=?

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

Question Number 132091 Answers: 2 Comments: 0

evaluate ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ∫_0 ^1 ((da.db.dc.dd.df)/(1−abcdf))

$${evaluate}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{{da}.{db}.{dc}.{dd}.{df}}{\mathrm{1}−{abcdf}} \\ $$

Question Number 132090 Answers: 3 Comments: 0

evaluate ∫_(−∞) ^∞ ((cosx)/((x^2 +1)^2 ))dx

$${evaluate}\:\int_{−\infty} ^{\infty} \frac{{cosx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 132089 Answers: 1 Comments: 0

∫_0 ^∞ (1/((x^4 −x^2 +1)^3 ))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{\left({x}^{\mathrm{4}} −{x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 132082 Answers: 2 Comments: 0

Solve ∫ (dx/((1+x^2 )^3 )) ?

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

Question Number 132070 Answers: 1 Comments: 0

I=∫ (x/((x^4 −1)^2 )) dx ?

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

Question Number 132023 Answers: 1 Comments: 0

.....nice calculus..... prove that::: 𝛗=∫_0 ^( ∞) ((cos(2x))/(cosh(x))) dx=^? (π/(2cosh(π)))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....{nice}\:\:{calculus}..... \\ $$$${prove}\:{that}::: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left(\mathrm{2}{x}\right)}{{cosh}\left({x}\right)}\:{dx}\overset{?} {=}\frac{\pi}{\mathrm{2}{cosh}\left(\pi\right)} \\ $$$$ \\ $$

Question Number 132010 Answers: 0 Comments: 1

Question Number 132004 Answers: 3 Comments: 0

very nice ∫ ((cosec^2 x−2019)/(cos^(2019) x)) dx ?

$$\:\mathrm{very}\:\mathrm{nice}\: \\ $$$$\:\int\:\frac{\mathrm{cosec}\:^{\mathrm{2}} \mathrm{x}−\mathrm{2019}}{\mathrm{cos}\:^{\mathrm{2019}} \mathrm{x}}\:\mathrm{dx}\:? \\ $$

Question Number 132000 Answers: 2 Comments: 0

super nice ∫_0 ^∞ ((1/((x^3 +(1/x^3 ))^2 )) )dx

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

Question Number 131991 Answers: 2 Comments: 0

...nice calculus... prove that : φ_1 =∫_0 ^( 1) li_2 (1−x^2 )=(π^2 /2) −4 φ_2 =∫_0 ^( 1) ((log(1−t))/(t^(3/4) (√(1−t))))dt=π^(3/2) .((√2)/(Γ^2 ((3/4))))(log(2)−(π/2)) hint 1:ψ((3/4))=_(easy) ^(why??) −γ+(π/2)−3log(2) hint 2 :ψ((1/2))=_(easy) ^(why??) −γ−2log(2)

$$\:\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:\:\:\:\:\:\:{calculus}...\:\: \\ $$$$\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\phi_{\mathrm{1}} =\int_{\mathrm{0}} ^{\:\mathrm{1}} {li}_{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)=\frac{\pi^{\mathrm{2}} }{\mathrm{2}}\:−\mathrm{4} \\ $$$$\:\:\:\:\phi_{\mathrm{2}} =\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{log}\left(\mathrm{1}−{t}\right)}{{t}^{\frac{\mathrm{3}}{\mathrm{4}}} \sqrt{\mathrm{1}−{t}}}{dt}=\pi^{\frac{\mathrm{3}}{\mathrm{2}}} .\frac{\sqrt{\mathrm{2}}}{\Gamma^{\mathrm{2}} \left(\frac{\mathrm{3}}{\mathrm{4}}\right)}\left({log}\left(\mathrm{2}\right)−\frac{\pi}{\mathrm{2}}\right) \\ $$$${hint}\:\:\:\mathrm{1}:\psi\left(\frac{\mathrm{3}}{\mathrm{4}}\right)\underset{{easy}} {\overset{{why}??} {=}}−\gamma+\frac{\pi}{\mathrm{2}}−\mathrm{3}{log}\left(\mathrm{2}\right) \\ $$$${hint}\:\:\mathrm{2}\::\psi\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\underset{{easy}} {\overset{{why}??} {=}}−\gamma−\mathrm{2}{log}\left(\mathrm{2}\right) \\ $$

Question Number 131974 Answers: 2 Comments: 0

Question Number 131969 Answers: 1 Comments: 0

... math analysis... φ= ∫_(−∞) ^( +∞) ((xsin(x))/(x^2 +2x+2))dx=? φ=∫_(−∞) ^( +∞) ((xsin(x))/((x+1)^2 +1))dx =^(x+1=t) ∫_(−∞) ^( +∞) (((t−1)sin(t−1))/(t^2 +1))dt =∫_(−∞) ^( +∞) ((tsin(t)cos(1)−tcos(t)sin(1)−sin(t)cos(1)+cos(t)sin(1))/(t^2 +1))dt =2cos(1)∫_0 ^( ∞) ((tsin(t))/(t^2 +1))dt+2sin(1)∫_0 ^( ∞) ((cos(t))/(t^2 +1))dt =2cos(1).(π/(2e))+2sin(1).(π/(2e)) =(π/e)(cos(1)+sin(1))....

$$\:\:\:\:\:\:\:\:\:\:\:\:...\:{math}\:\:{analysis}... \\ $$$$\:\:\:\:\phi=\:\:\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}{dx}=? \\ $$$$\:\:\:\:\:\phi=\int_{−\infty} ^{\:+\infty} \frac{{xsin}\left({x}\right)}{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{1}}{dx} \\ $$$$\:\:\:\:\:\:\:\overset{{x}+\mathrm{1}={t}} {=}\int_{−\infty} ^{\:+\infty} \frac{\left({t}−\mathrm{1}\right){sin}\left({t}−\mathrm{1}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}{dt} \\ $$$$\:\:\:\:\:\:\:\:=\int_{−\infty} ^{\:+\infty} \frac{{tsin}\left({t}\right){cos}\left(\mathrm{1}\right)−{tcos}\left({t}\right){sin}\left(\mathrm{1}\right)−{sin}\left({t}\right){cos}\left(\mathrm{1}\right)+{cos}\left({t}\right){sin}\left(\mathrm{1}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}{dt} \\ $$$$=\mathrm{2}{cos}\left(\mathrm{1}\right)\int_{\mathrm{0}} ^{\:\infty} \frac{{tsin}\left({t}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}{dt}+\mathrm{2}{sin}\left(\mathrm{1}\right)\int_{\mathrm{0}} ^{\:\infty} \frac{{cos}\left({t}\right)}{{t}^{\mathrm{2}} +\mathrm{1}}{dt} \\ $$$$=\mathrm{2}{cos}\left(\mathrm{1}\right).\frac{\pi}{\mathrm{2}{e}}+\mathrm{2}{sin}\left(\mathrm{1}\right).\frac{\pi}{\mathrm{2}{e}} \\ $$$$\:=\frac{\pi}{{e}}\left({cos}\left(\mathrm{1}\right)+{sin}\left(\mathrm{1}\right)\right).... \\ $$$$\:\:\: \\ $$

Question Number 131961 Answers: 2 Comments: 0

∫(sin^4 x.cos^4 x)dx

$$\int\left({sin}^{\mathrm{4}} {x}.{cos}^{\mathrm{4}} {x}\right){dx} \\ $$

Question Number 131957 Answers: 2 Comments: 0

Evaluate ∫_(−∞) ^∞ ((sinx)/(x^2 +2x+2))dx

$${Evaluate}\:\:\int_{−\infty} ^{\infty} \frac{{sinx}}{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}{dx} \\ $$

Question Number 131919 Answers: 1 Comments: 0

Question Number 131891 Answers: 0 Comments: 0

∫ ((x!)/( (√x))) dx = ?

$$\int\:\frac{{x}!}{\:\sqrt{{x}}}\:{dx}\:=\:? \\ $$

Question Number 131875 Answers: 1 Comments: 0

...nice calculus.. Φ=∫_0 ^( ∞) ((ln(cosh(x)))/(cosh(x)))dx=???

$$\:\:\:\:\:\:\:\:\:\:\:...{nice}\:\:\:{calculus}.. \\ $$$$\:\:\:\:\:\Phi=\int_{\mathrm{0}} ^{\:\infty} \frac{{ln}\left({cosh}\left({x}\right)\right)}{{cosh}\left({x}\right)}{dx}=??? \\ $$

Question Number 131874 Answers: 0 Comments: 0

... calculus ... φ =∫_0 ^( ∞) ((tanh^2 (x)dx)/x^2 ) =?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:{calculus}\:... \\ $$$$\:\:\:\phi\:=\int_{\mathrm{0}} ^{\:\infty} \frac{{tanh}^{\mathrm{2}} \left({x}\right){dx}}{{x}^{\mathrm{2}} }\:=? \\ $$$$ \\ $$

Question Number 131866 Answers: 2 Comments: 0

...advanced calculus... Ω=∫_0 ^( ∞) (dx/(x^5 (e^(1/x) −1)))=?

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:\:{calculus}... \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{{dx}}{{x}^{\mathrm{5}} \left({e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}\right)}=? \\ $$$$ \\ $$

Question Number 131852 Answers: 1 Comments: 0

... analysis (II)... evaluate :: ∅=∫_1 ^( 10) x^2 d({x})=? {x} :: fractional part of x ...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...\:\:{analysis}\:\left({II}\right)... \\ $$$$\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\varnothing=\int_{\mathrm{1}} ^{\:\mathrm{10}} {x}^{\mathrm{2}} {d}\left(\left\{{x}\right\}\right)=? \\ $$$$\:\:\:\:\:\:\:\left\{{x}\right\}\:::\:{fractional}\:{part}\:{of}\:{x}\:... \\ $$$$ \\ $$

Question Number 131849 Answers: 3 Comments: 0

∗∗∗ calculus (I) ∗∗∗ please evaluate:: φ=∫(dx/(sin(2x)ln(tan(x)))) Trinity College Cambridge ....1897...

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\ast\ast\ast\:\:\:{calculus}\:\left({I}\right)\:\ast\ast\ast \\ $$$$\:\:\:{please}\:\:{evaluate}:: \\ $$$$\:\:\:\:\:\:\:\:\phi=\int\frac{{dx}}{{sin}\left(\mathrm{2}{x}\right){ln}\left({tan}\left({x}\right)\right)} \\ $$$$\:\:\:\:\:\:{Trinity}\:{College} \\ $$$$\:\:\:\:\:\:\:{Cambridge}\:....\mathrm{1897}... \\ $$

Question Number 131816 Answers: 0 Comments: 0

calculate ∫_(−∞) ^∞ ((cos(x^n ))/(1+x^n ))dx with n≥2 integr

$$\mathrm{calculate}\:\int_{−\infty} ^{\infty} \:\frac{\mathrm{cos}\left(\mathrm{x}^{\mathrm{n}} \right)}{\mathrm{1}+\mathrm{x}^{\mathrm{n}} }\mathrm{dx}\:\:\mathrm{with}\:\mathrm{n}\geqslant\mathrm{2}\:\:\mathrm{integr} \\ $$

Question Number 131810 Answers: 1 Comments: 0

f(x)= { ((−2x ; x≤0)),((f(x−1) ; x>0)) :} ∫_0 ^(100) f(x)dx =?

$${f}\left({x}\right)=\begin{cases}{−\mathrm{2}{x}\:\:\:\:\:\:\:\:\:\:;\:\:{x}\leqslant\mathrm{0}}\\{{f}\left({x}−\mathrm{1}\right)\:\:\:;\:\:{x}>\mathrm{0}}\end{cases} \\ $$$$ \\ $$$$\:\:\underset{\mathrm{0}} {\overset{\mathrm{100}} {\int}}{f}\left({x}\right){dx}\:=? \\ $$

Question Number 131807 Answers: 1 Comments: 1

f(x)=2−x ∫_0 ^1 f(x)dx ⇒ ∫_0 ^1 f(x)dx=?

$${f}\left({x}\right)=\mathrm{2}−{x}\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({x}\right){dx}\:\:\Rightarrow\:\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({x}\right){dx}=? \\ $$$$ \\ $$$$ \\ $$

Pg 102 Pg 103 Pg 104 Pg 105 Pg 106 Pg 107 Pg 108 Pg 109 Pg 110 Pg 111

Terms of Service

Contact: info@tinkutara.com