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IntegrationQuestion and Answers: Page 19

Question Number 190937 Answers: 1 Comments: 0

Show that ∫ ((sech (√x) tanh (√x))/( (√x)))=−(2/(cosh (√x)))

$$\mathrm{Show}\:\:\mathrm{that}\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\:\:\frac{\mathrm{sech}\:\sqrt{\mathrm{x}}\:\mathrm{tanh}\:\sqrt{\mathrm{x}}}{\:\sqrt{\mathrm{x}}}=−\frac{\mathrm{2}}{\mathrm{cosh}\:\sqrt{\mathrm{x}}} \\ $$

Question Number 190841 Answers: 1 Comments: 0

Question Number 190812 Answers: 1 Comments: 1

Question Number 190809 Answers: 0 Comments: 2

Question Number 190754 Answers: 1 Comments: 0

I = ∫_0 ^( π) e^(acos t) cos (asin t)dt

$$\:\:\:\:\:\:{I}\:\:\:=\:\:\:\:\int_{\mathrm{0}} ^{\:\pi} {e}^{{a}\mathrm{cos}\:{t}} \:\mathrm{cos}\:\left({a}\mathrm{sin}\:\:{t}\right){dt} \\ $$

Question Number 190740 Answers: 1 Comments: 0

∫(dx/((x^2 +5)^2 ))=?

$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{5}\right)^{\mathrm{2}} }=? \\ $$

Question Number 190708 Answers: 1 Comments: 0

Question Number 190700 Answers: 1 Comments: 0

Question Number 190692 Answers: 1 Comments: 0

calculate 𝛗 = ∫_0 ^( ∞) (( sin^( 3) (x ) ln( x ))/x) dx = ? @ nice − mathematics

$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{calculate}\: \\ $$$$\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\mathrm{sin}^{\:\mathrm{3}} \left({x}\:\right)\:\mathrm{ln}\left(\:{x}\:\right)}{{x}}\:\mathrm{d}{x}\:=\:?\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:@\:\mathrm{nice}\:−\:\mathrm{mathematics}\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 190624 Answers: 2 Comments: 0

Question Number 190623 Answers: 2 Comments: 0

calculate : Σ_(n=1) ^∞ (( (−1)^( n−1) )/n) cos ((( nπ)/3) ) =?

$$ \\ $$$$\:\:\:\:\:\:\mathrm{calculate}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\left(−\mathrm{1}\right)^{\:{n}−\mathrm{1}} }{{n}}\:\mathrm{cos}\:\left(\frac{\:{n}\pi}{\mathrm{3}}\:\right)\:=? \\ $$$$ \\ $$

Question Number 190622 Answers: 1 Comments: 0

prove : ∫_(−∞) ^( ∞) ((( x)/( ⋮ )^2 dx= Σ_(k=1) ^∞ (1/( k^2 )) ⋖))

$$ \\ $$$$\:\:\:{prove}\:: \\ $$$$\:\:\int_{−\infty} ^{\:\infty} \:\:\:\left(\frac{\:{x}}{\left.\:\underline{\vdots} \right)^{\mathrm{2}} \mathrm{d}{x}=\:\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\:{k}\:^{\mathrm{2}} }\:\:\:\lessdot}\right. \\ $$

Question Number 190552 Answers: 1 Comments: 1

∫_(1/2) ^2 ln(((ln(x+(1/x)))/(ln(x^2 −x+((17)/6)))))dx=?

$$\int_{\frac{\mathrm{1}}{\mathrm{2}}} ^{\mathrm{2}} {ln}\left(\frac{{ln}\left({x}+\frac{\mathrm{1}}{{x}}\right)}{{ln}\left({x}^{\mathrm{2}} −{x}+\frac{\mathrm{17}}{\mathrm{6}}\right)}\right){dx}=? \\ $$

Question Number 190533 Answers: 1 Comments: 0

Question Number 190487 Answers: 1 Comments: 0

Question Number 190419 Answers: 1 Comments: 0

∫_(−1) ^1 ∫_(−(√(1−y^2 ))) ^(√(1−y^2 )) ln (x^2 +y^2 +1)dx dy =?

$$\:\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\underset{−\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\overset{\sqrt{\mathrm{1}−{y}^{\mathrm{2}} }} {\int}}\:\mathrm{ln}\:\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{1}\right){dx}\:{dy}\:=? \\ $$

Question Number 190385 Answers: 1 Comments: 0

Question Number 190371 Answers: 1 Comments: 0

Question Number 190360 Answers: 3 Comments: 0

if x+ (1/x) = ϕ ( Golden ratio) ⇒ x^( 2000) + (1/x^( 2000) )=?

$$ \\ $$$${if}\:\:\:{x}+\:\frac{\mathrm{1}}{{x}}\:=\:\varphi\:\left(\:\:{Golden}\:{ratio}\right) \\ $$$$\:\:\:\:\:\Rightarrow\:\:\:{x}^{\:\mathrm{2000}} +\:\frac{\mathrm{1}}{{x}^{\:\mathrm{2000}} }=? \\ $$$$ \\ $$$$ \\ $$

Question Number 190318 Answers: 2 Comments: 0

Question Number 190172 Answers: 1 Comments: 0

Integrate ∫_0 ^1 Sin^2 (2Πx)dx

$${Integrate}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \:{Sin}^{\mathrm{2}} \left(\mathrm{2}\Pi{x}\right){dx} \\ $$

Question Number 190168 Answers: 0 Comments: 2

1)∫^∞ _0 ((sin x)/(x^p +sin x))dx ,p>0 2)∫^∞ _π ((xcos x)/(x^p +x^q ))dx,p>0and q>0 3)∫^∞ _0 ((sin x^p )/( x^q ))dx, p>0,q>0 4)∫^2 _0 (dx/(∣ln x∣^p )) ,p>0 5)∫^1 _0 ((cos(1/(1−x)))/( ((1−x^2 ))^(1/n) ))dx 6)∫^∞ _0 (dx/(x^p ((sin^2 x))^(1/3) ))

$$\left.\mathrm{1}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{sin}\:{x}}{{x}^{{p}} +{sin}\:{x}}{dx}\:,{p}>\mathrm{0} \\ $$$$\left.\mathrm{2}\right)\underset{\pi} {\int}^{\infty} \frac{{xcos}\:{x}}{{x}^{{p}} +{x}^{{q}} }{dx},{p}>\mathrm{0}{and}\:{q}>\mathrm{0} \\ $$$$\left.\mathrm{3}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{sin}\:{x}^{{p}} }{\:{x}^{{q}} }{dx},\:{p}>\mathrm{0},{q}>\mathrm{0} \\ $$$$\left.\mathrm{4}\right)\underset{\mathrm{0}} {\int}^{\mathrm{2}} \frac{{dx}}{\mid{ln}\:{x}\mid^{{p}} }\:,{p}>\mathrm{0} \\ $$$$\left.\mathrm{5}\right)\underset{\mathrm{0}} {\int}^{\mathrm{1}} \frac{{cos}\frac{\mathrm{1}}{\mathrm{1}−{x}}}{\:\sqrt[{{n}}]{\mathrm{1}−{x}^{\mathrm{2}} }}{dx} \\ $$$$\left.\mathrm{6}\right)\underset{\mathrm{0}} {\int}^{\infty} \frac{{dx}}{{x}^{{p}} \sqrt[{\mathrm{3}}]{{sin}^{\mathrm{2}} {x}}} \\ $$

Question Number 190009 Answers: 0 Comments: 2

Question Number 189949 Answers: 1 Comments: 0

Question Number 189890 Answers: 0 Comments: 0

Question Number 189825 Answers: 2 Comments: 0

∫^b _a (√((x−a)(b−x)))=¿

$$\underset{{a}} {\int}^{{b}} \sqrt{\left({x}−{a}\right)\left({b}−{x}\right)}=¿ \\ $$

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