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IntegrationQuestion and Answers: Page 77

Question Number 142116 Answers: 2 Comments: 0

use trigonometric substitution to solve ∫(x^3 /( (√(9−x^2 ))))dx

$${use}\:{trigonometric}\:{substitution}\:{to}\:{solve} \\ $$$$\int\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{9}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 142060 Answers: 2 Comments: 0

Question Number 142028 Answers: 2 Comments: 0

............Calculus......... ∫_0 ^( ∞) ((sin(sin(x)).e^(cos(x)) )/x)dx=??? ............m.n.....

$$\:\:\:\:\:\:\:\:............{Calculus}.........\: \\ $$$$\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}\left({sin}\left({x}\right)\right).{e}^{{cos}\left({x}\right)} }{{x}}{dx}=??? \\ $$$$\:............{m}.{n}..... \\ $$

Question Number 142103 Answers: 2 Comments: 0

∫_(−∞) ^∞ ((tan^(−1) ((√(x^2 +2))))/((x^2 +1)(√(x^2 +2))))dx

$$\int_{−\infty} ^{\infty} \frac{{tan}^{−\mathrm{1}} \left(\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}\right)}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\sqrt{{x}^{\mathrm{2}} +\mathrm{2}}}{dx} \\ $$

Question Number 141998 Answers: 2 Comments: 0

∫_0 ^(π/2) ((sin (40x))/(sin (5x))) dx

$$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{\mathrm{sin}\:\left(\mathrm{40}{x}\right)}{\mathrm{sin}\:\left(\mathrm{5}{x}\right)}\:{dx}\: \\ $$

Question Number 141946 Answers: 1 Comments: 0

....nice calculus... lim_(n→∞) n∫_0 ^( 1) (((2x)/(1+x)))^n =???

$$\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:{calculus}... \\ $$$$\:\:\:\:{lim}_{{n}\rightarrow\infty} {n}\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}}\right)^{{n}} =??? \\ $$

Question Number 141944 Answers: 4 Comments: 0

∫x^2 (√(9x^2 +25))dx

$$\int{x}^{\mathrm{2}} \sqrt{\mathrm{9}{x}^{\mathrm{2}} +\mathrm{25}}{dx} \\ $$

Question Number 141943 Answers: 2 Comments: 0

∫(dx/(x(√(16−4x^2 ))))

$$\int\frac{{dx}}{{x}\sqrt{\mathrm{16}−\mathrm{4}{x}^{\mathrm{2}} }} \\ $$

Question Number 141931 Answers: 2 Comments: 0

calculate ∫_0 ^∞ e^(−2x) ln(1+e^(3x) )dx

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

Question Number 141930 Answers: 0 Comments: 0

calculate ∫_0 ^∞ e^(−(x^3 +(1/x^3 ))) dx

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

Question Number 141929 Answers: 1 Comments: 0

find ∫_0 ^∞ e^(−(t^2 +(1/t^2 ))) dt

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

Question Number 141868 Answers: 1 Comments: 1

......Advanced .....Calculus........ ..... ∫_( R) ^ (e^(−x^2 ) /((1+x^2 )^2 )) dx=?

$$\:\:\:\:\:\:\:\:\:\:\:\:......\mathscr{A}{dvanced}\:\:.....\mathscr{C}{alculus}........ \\ $$$$\:\:\:\:\:\:\:\:\:.....\:\:\int_{\:\mathbb{R}} ^{\:} \frac{{e}^{−{x}^{\mathrm{2}} } }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\:{dx}=? \\ $$$$\:\:\:\:\: \\ $$

Question Number 141859 Answers: 2 Comments: 1

∫_0 ^∞ ((4e^(−x^2 ) )/((2x^2 +1)^2 )) dx

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

Question Number 141848 Answers: 4 Comments: 0

I = ∫ (1/( (√(1−x^2 ))−1)) dx

$$\:\mathscr{I}\:=\:\int\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }−\mathrm{1}}\:{dx}\: \\ $$

Question Number 141847 Answers: 2 Comments: 0

I = ∫ ((sec x)/(1+csc x)) dx

$$\:\mathcal{I}\:=\:\int\:\frac{\mathrm{sec}\:{x}}{\mathrm{1}+\mathrm{csc}\:{x}}\:{dx}\: \\ $$

Question Number 142255 Answers: 1 Comments: 0

find the value of ∫_0 ^1 (√(x^4 −5x^2 +4)) dx? solution please.

$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{x}^{\mathrm{4}} −\mathrm{5x}^{\mathrm{2}} +\mathrm{4}}\:\mathrm{dx}? \\ $$$$\mathrm{solution}\:\mathrm{please}. \\ $$

Question Number 142256 Answers: 1 Comments: 0

∫_0 ^∞ ((sinx)/x^(1−a) )dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{sinx}}{{x}^{\mathrm{1}−{a}} }{dx} \\ $$

Question Number 141811 Answers: 1 Comments: 0

Θ:=(Σ_(n=1) ^∞ (n^4 /(2^n . n!)))^(1/2) =?

$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\Theta:=\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{4}} }{\mathrm{2}^{{n}} \:.\:{n}!}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} =? \\ $$

Question Number 141757 Answers: 0 Comments: 1

Γ(n+(1/2))=(((√π)∙Γ(2n+1))/(2^(2n) Γ(n+1)))

$$\Gamma\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)=\frac{\sqrt{\pi}\centerdot\Gamma\left(\mathrm{2n}+\mathrm{1}\right)}{\mathrm{2}^{\mathrm{2n}} \Gamma\left(\mathrm{n}+\mathrm{1}\right)} \\ $$

Question Number 141755 Answers: 0 Comments: 0

Question Number 141719 Answers: 2 Comments: 0

∫_0 ^∞ (e^(−x^2 ) /((x^2 +(1/2))^2 ))dx=?

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

Question Number 141713 Answers: 1 Comments: 0

Question Number 141691 Answers: 1 Comments: 0

....Calculus(I).... 𝛗:=∫_(1/(2 )) ^( 1) (1/(x^2 (1+x^4 )^(3/4) ))dx=???

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....{Calculus}\left({I}\right).... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\frac{\mathrm{1}}{\mathrm{2}\:}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{{x}^{\mathrm{2}} \left(\mathrm{1}+{x}^{\mathrm{4}} \right)^{\frac{\mathrm{3}}{\mathrm{4}}} }{dx}=??? \\ $$

Question Number 141685 Answers: 1 Comments: 0

......nice ... ... ... calculus..... If lim_(x→0) ((tan(x))/x) = 1 , prove that: lim(1/x)((1/x)−(1/(tan(x))))=(1/3)

$$\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:......{nice}\:...\:...\:...\:{calculus}..... \\ $$$$\:\:\mathrm{I}{f}\:\:{lim}_{{x}\rightarrow\mathrm{0}} \frac{{tan}\left({x}\right)}{{x}}\:=\:\mathrm{1}\:,\:{prove}\:{that}: \\ $$$$\:\:\:\:\:\:\:\:{lim}\frac{\mathrm{1}}{{x}}\left(\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{{tan}\left({x}\right)}\right)=\frac{\mathrm{1}}{\mathrm{3}} \\ $$

Question Number 143167 Answers: 2 Comments: 0

∫arctan((√((√x)+1)))dx=??? propose′ par Rodrigue

$$\int\boldsymbol{{arctan}}\left(\sqrt{\sqrt{\boldsymbol{{x}}}+\mathrm{1}}\right)\boldsymbol{{dx}}=??? \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{propose}}'\:\boldsymbol{{par}}\:\boldsymbol{{Rodrigue}} \\ $$

Question Number 141649 Answers: 1 Comments: 0

.......advanced calculus...... prove that−:: φ:=∫_0 ^( ∞) ((cos(2πx^2 ))/(cosh^2 (πx)))dx=(1/4) ✓

$$\:\:\:\:\:\:\:\:\:.......{advanced}\:\:{calculus}...... \\ $$$$\:\:\:\:{prove}\:\:{that}−:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\phi:=\int_{\mathrm{0}} ^{\:\infty} \:\frac{{cos}\left(\mathrm{2}\pi{x}^{\mathrm{2}} \right)}{{cosh}^{\mathrm{2}} \left(\pi{x}\right)}{dx}=\frac{\mathrm{1}}{\mathrm{4}}\:\:\checkmark \\ $$

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