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IntegrationQuestion and Answers: Page 162

Question Number 96076 Answers: 2 Comments: 0

∫ 3x.2^x dx ?

$$\int\:\mathrm{3x}.\mathrm{2}^{\mathrm{x}} \:\mathrm{dx}\:?\: \\ $$

Question Number 96034 Answers: 2 Comments: 1

∫_0 ^∞ (1/(x^(10) +1))dx=((2π)/(5((√5)−1)))=((πφ)/5)

$$\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{1}}{{x}^{\mathrm{10}} +\mathrm{1}}{dx}=\frac{\mathrm{2}\pi}{\mathrm{5}\left(\sqrt{\mathrm{5}}−\mathrm{1}\right)}=\frac{\pi\phi}{\mathrm{5}} \\ $$

Question Number 95951 Answers: 4 Comments: 3

Question Number 95949 Answers: 2 Comments: 0

∫_0 ^(+∞) (x^2 /(e^x^2 −1))dx

$$\int_{\mathrm{0}} ^{+\infty} \frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{e}^{\mathrm{x}^{\mathrm{2}} } −\mathrm{1}}\mathrm{dx} \\ $$

Question Number 95943 Answers: 0 Comments: 0

f is a integrable function wich verify f(x+π)=f(x) prove that ∫_0 ^∞ f(x)×((sinx)/x)dx =∫_0 ^(π/2) f(x)dx

$$\mathrm{f}\:\mathrm{is}\:\mathrm{a}\:\mathrm{integrable}\:\mathrm{function}\:\mathrm{wich}\:\mathrm{verify}\:\mathrm{f}\left(\mathrm{x}+\pi\right)=\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{prove}\:\mathrm{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \:\mathrm{f}\left(\mathrm{x}\right)×\frac{\mathrm{sinx}}{\mathrm{x}}\mathrm{dx}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$

Question Number 95848 Answers: 4 Comments: 0

∫_0 ^(π/2) (dx/(√(1+sin x))) ?

$$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{dx}}{\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}}\:?\: \\ $$

Question Number 95845 Answers: 1 Comments: 0

calculate ∫_0 ^1 (−1)^([(2/x)]) dx

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\left(−\mathrm{1}\right)^{\left[\frac{\mathrm{2}}{\mathrm{x}}\right]} \:\mathrm{dx} \\ $$

Question Number 95844 Answers: 2 Comments: 0

cacuate ∫_(−(π/4)) ^(π/4) ln(1+a cos^2 t)dt with ∣a∣<1

$$\mathrm{cacuate}\:\:\int_{−\frac{\pi}{\mathrm{4}}} ^{\frac{\pi}{\mathrm{4}}} \mathrm{ln}\left(\mathrm{1}+\mathrm{a}\:\mathrm{cos}^{\mathrm{2}} \mathrm{t}\right)\mathrm{dt}\:\mathrm{with}\:\mid\mathrm{a}\mid<\mathrm{1} \\ $$

Question Number 95800 Answers: 1 Comments: 2

if f(0)=1 f(1)=2 and ∫_0 ^1 f(x) dx=3 than ∫_0 ^1 x f(x) dx = ? a. 1 b. −1 c. 2 d. −2

$${if}\:\:{f}\left(\mathrm{0}\right)=\mathrm{1} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{2}\:{and}\:\int_{\mathrm{0}} ^{\mathrm{1}} {f}\left({x}\right)\:{dx}=\mathrm{3}\: \\ $$$${than}\:\int_{\mathrm{0}} ^{\mathrm{1}} {x}\:{f}\left({x}\right)\:{dx}\:=\:? \\ $$$$ \\ $$$${a}.\:\mathrm{1} \\ $$$${b}.\:−\mathrm{1} \\ $$$${c}.\:\mathrm{2} \\ $$$${d}.\:−\mathrm{2} \\ $$$$ \\ $$

Question Number 95763 Answers: 0 Comments: 3

∫ x (√(x^3 +1)) dx ?

$$\int\:{x}\:\sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}\:? \\ $$

Question Number 95738 Answers: 1 Comments: 2

∫ ((p−tan x)/(p+tan x)) dx

$$\int\:\frac{\mathrm{p}−\mathrm{tan}\:\mathrm{x}}{\mathrm{p}+\mathrm{tan}\:\mathrm{x}}\:\mathrm{dx}\: \\ $$

Question Number 95722 Answers: 2 Comments: 0

use cylinder ring method y = 2x−1 y = −2x + 3 x = 2 y−axis

$${use}\:{cylinder}\:{ring}\:{method} \\ $$$$ \\ $$$${y}\:=\:\mathrm{2}{x}−\mathrm{1} \\ $$$${y}\:=\:−\mathrm{2}{x}\:+\:\mathrm{3} \\ $$$${x}\:=\:\mathrm{2}\: \\ $$$$ \\ $$$${y}−{axis}\: \\ $$$$ \\ $$$$ \\ $$

Question Number 95692 Answers: 1 Comments: 0

calculate ∫_(−∞) ^∞ ((xsin(2x))/((x^2 +x+1)^2 ))dx

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

Question Number 95691 Answers: 0 Comments: 1

calculate ∫ ((x+1)/(√((x+3)(2−x))))dx

$$\mathrm{calculate}\:\int\:\:\frac{\mathrm{x}+\mathrm{1}}{\sqrt{\left(\mathrm{x}+\mathrm{3}\right)\left(\mathrm{2}−\mathrm{x}\right)}}\mathrm{dx} \\ $$

Question Number 95690 Answers: 0 Comments: 0

calculate I =∫_0 ^(π/2) cos^3 (x)sh^2 (x)dx and J =∫_0 ^(π/3) sin^3 x ch^2 x

$$\mathrm{calculate}\:\mathrm{I}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{3}} \left(\mathrm{x}\right)\mathrm{sh}^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{dx}\:\mathrm{and}\:\mathrm{J}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} \:\mathrm{sin}^{\mathrm{3}} \mathrm{x}\:\mathrm{ch}^{\mathrm{2}} \mathrm{x} \\ $$

Question Number 95673 Answers: 0 Comments: 2

∫_0 ^1 (1/((√(3 + 4x−4x^2 )) )) dx = ?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\sqrt{\mathrm{3}\:+\:\mathrm{4}{x}−\mathrm{4}{x}^{\mathrm{2}} }\:}\:{dx}\:=\:? \\ $$

Question Number 95653 Answers: 2 Comments: 0

arc length 3x^(3/2) −1 from x=0 and x=1 help please sir

$${arc}\:{length}\:\mathrm{3}{x}^{\frac{\mathrm{3}}{\mathrm{2}}} −\mathrm{1}\: \\ $$$${from}\:{x}=\mathrm{0}\:{and}\:{x}=\mathrm{1}\: \\ $$$${help}\:{please}\:{sir} \\ $$

Question Number 95650 Answers: 1 Comments: 0

∫_0 ^1 {(−1)^(⌊(1/x)⌋) (1/x)}dx {..}is fractional part ⌊..⌋ is floor function

$$\int_{\mathrm{0}} ^{\mathrm{1}} \left\{\left(−\mathrm{1}\right)^{\lfloor\frac{\mathrm{1}}{{x}}\rfloor} \frac{\mathrm{1}}{{x}}\right\}{dx} \\ $$$$\left\{..\right\}{is}\:{fractional}\:{part} \\ $$$$\lfloor..\rfloor\:{is}\:{floor}\:{function} \\ $$

Question Number 95624 Answers: 1 Comments: 0

∫_1 ^5 x(∫f(x)dx) dx = 24 ∫_1 ^3 (2−f(x))dx ?

$$\overset{\mathrm{5}} {\int}_{\mathrm{1}} {x}\left(\int\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\right)\:{dx}\:=\:\mathrm{24} \\ $$$$\overset{\mathrm{3}} {\int}_{\mathrm{1}} \left(\mathrm{2}−\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}\:?\: \\ $$

Question Number 95593 Answers: 0 Comments: 2

if f(x) = ((sin x)/((1 + x^2 +x^6 )^2 )) is odd, find ∫_(−3) ^3 f(x) dx

$$\mathrm{if}\:{f}\left({x}\right)\:=\:\frac{\mathrm{sin}\:{x}}{\left(\mathrm{1}\:+\:{x}^{\mathrm{2}} \:+{x}^{\mathrm{6}} \right)^{\mathrm{2}} }\:\mathrm{is}\:\mathrm{odd}, \\ $$$$\mathrm{find}\:\underset{−\mathrm{3}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right)\:{dx} \\ $$

Question Number 95586 Answers: 1 Comments: 1

find ∫ (dx/(x^n (x+1)^m )) m and n integr

$${find}\:\int\:\:\frac{{dx}}{{x}^{{n}} \left({x}+\mathrm{1}\right)^{{m}} }\:\: \\ $$$${m}\:{and}\:{n}\:{integr} \\ $$

Question Number 95585 Answers: 1 Comments: 0

calculate ∫_2 ^(+∞) (dx/((x−1)^4 (x^2 +x+1)^2 ))

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

Question Number 95584 Answers: 3 Comments: 0

calculate ∫_1 ^(+∞) (dx/(x^3 (2x+1)^4 )) 1)without use of decomposition 2)by use of decomposition

$${calculate}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{dx}}{{x}^{\mathrm{3}} \left(\mathrm{2}{x}+\mathrm{1}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right){without}\:{use}\:{of}\:{decomposition} \\ $$$$\left.\mathrm{2}\right){by}\:{use}\:{of}\:{decomposition} \\ $$

Question Number 95581 Answers: 0 Comments: 0

∫ln∣cot(x/2)∣dx

$$\int\mathrm{ln}\mid\mathrm{cot}\left(\mathrm{x}/\mathrm{2}\right)\mid\mathrm{dx} \\ $$

Question Number 95563 Answers: 1 Comments: 0

∫((ax^2 +bx+c)/((x−p)(x−q)(x−r)))dx

$$\int\frac{\mathrm{ax}^{\mathrm{2}} +\mathrm{bx}+\mathrm{c}}{\left(\mathrm{x}−\mathrm{p}\right)\left(\mathrm{x}−\mathrm{q}\right)\left(\mathrm{x}−\mathrm{r}\right)}\mathrm{dx} \\ $$

Question Number 95562 Answers: 1 Comments: 0

show that ∫((sin (x−θ))/(sin x))dx=xcos x−sin θlog sin x

$$\mathrm{show}\:\mathrm{that} \\ $$$$\int\frac{\mathrm{sin}\:\left(\mathrm{x}−\theta\right)}{\mathrm{sin}\:\mathrm{x}}\mathrm{dx}=\mathrm{xcos}\:\mathrm{x}−\mathrm{sin}\:\theta\mathrm{log}\:\mathrm{sin}\:\mathrm{x} \\ $$

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