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IntegrationQuestion and Answers: Page 170

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

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

Question Number 95549 Answers: 3 Comments: 0

∫_(−π) ^π ∣cos^3 x∣dx

$$\int_{−\pi} ^{\pi} \mid{cos}^{\mathrm{3}} {x}\mid{dx} \\ $$

Question Number 95548 Answers: 0 Comments: 2

Question Number 95547 Answers: 2 Comments: 0

∫((2x^3 dx)/(2x^2 −4x+3))=?

$$\int\frac{\mathrm{2}{x}^{\mathrm{3}} {dx}}{\mathrm{2}{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{3}}=? \\ $$

Question Number 95546 Answers: 2 Comments: 0

∫x^2 (√(a^2 +x^2 ))dx=?

$$\int{x}^{\mathrm{2}} \sqrt{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }{dx}=? \\ $$

Question Number 95518 Answers: 0 Comments: 0

Question Number 95494 Answers: 0 Comments: 0

Question Number 95592 Answers: 0 Comments: 1

∫ ((sin 2x)/(sin^4 x + cos^4 x)) dx

$$\int\:\frac{\mathrm{sin}\:\mathrm{2x}}{\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\:+\:\mathrm{cos}\:^{\mathrm{4}} \mathrm{x}}\:\mathrm{dx} \\ $$

Question Number 95449 Answers: 1 Comments: 7

∫∫ (√(x^2 +y^2 )) dxdy = where D : x^2 +y^2 ≤ 100

$$\int\int\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} }\:\mathrm{dxdy}\:=\: \\ $$$$\mathrm{where}\:\mathrm{D}\::\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \:\leqslant\:\mathrm{100}\: \\ $$

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