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IntegrationQuestion and Answers: Page 149

Question Number 103515 Answers: 1 Comments: 0

given f(x) = f(x+(π/6)) ∀x∈R if ∫_0 ^(π/6) f(x)dx = T then ∫_π ^(7π/3) f(x+π) dx ?

$${given}\:{f}\left({x}\right)\:=\:{f}\left({x}+\frac{\pi}{\mathrm{6}}\right)\:\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{6}} {\int}}{f}\left({x}\right){dx}\:=\:{T}\:{then}\:\underset{\pi} {\overset{\mathrm{7}\pi/\mathrm{3}} {\int}}{f}\left({x}+\pi\right) \\ $$$${dx}\:? \\ $$

Question Number 103512 Answers: 2 Comments: 2

∫ ((x dx)/((cot x+tan x)^2 )) = (a) (x/(16))−((x sin 4x)/(32))−((cos 4x)/(128))+c (b) (x/(16))+((x sin 4x)/(32))−((cos 4x)/(128))+c (c) (x/(16))+((xsin 4x)/(64))+((cos 4x)/(128))+c (d)(x/(16))+((xcos 4x)/(32))+((sin 4x)/(128))+c

$$\int\:\frac{{x}\:{dx}}{\left(\mathrm{cot}\:{x}+\mathrm{tan}\:{x}\right)^{\mathrm{2}} }\:= \\ $$$$\left({a}\right)\:\frac{{x}}{\mathrm{16}}−\frac{{x}\:\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{32}}−\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({b}\right)\:\frac{{x}}{\mathrm{16}}+\frac{{x}\:\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{32}}−\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({c}\right)\:\frac{{x}}{\mathrm{16}}+\frac{{x}\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{64}}+\frac{\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$$$\left({d}\right)\frac{{x}}{\mathrm{16}}+\frac{{x}\mathrm{cos}\:\mathrm{4}{x}}{\mathrm{32}}+\frac{\mathrm{sin}\:\mathrm{4}{x}}{\mathrm{128}}+{c} \\ $$

Question Number 103511 Answers: 1 Comments: 1

∫ (dx/((√x) ((x)^(1/4) +1))) =__ (a) −((9 (x)^(1/4) +1)/(18((x)^(1/4) +1)^9 )) + c (b) ((9 (x)^(1/4) +1)/(18((x)^(1/4) +1)^9 )) +c (c) −((9 (x)^(1/4) −1)/(18((x)^(1/4) +1)^9 )) +c (d) ((9 (x)^(1/4) +1)/(8((x)^(1/4) +1)^9 )) + c

$$\int\:\frac{{dx}}{\sqrt{{x}}\:\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)}\:=\_\_ \\ $$$$\left({a}\right)\:−\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+\:{c}\: \\ $$$$\left({b}\right)\:\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}+\mathrm{1}\right)^{\mathrm{9}} }\:+{c} \\ $$$$\left({c}\right)\:−\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}\:−\mathrm{1}}{\mathrm{18}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+{c} \\ $$$$\left({d}\right)\:\frac{\mathrm{9}\:\sqrt[{\mathrm{4}}]{{x}}+\mathrm{1}}{\mathrm{8}\left(\sqrt[{\mathrm{4}}]{{x}}\:+\mathrm{1}\right)^{\mathrm{9}} }\:+\:{c} \\ $$

Question Number 103537 Answers: 1 Comments: 0

∫(x/((a^2 cosx+b^2 sinx)))dx

$$\int\frac{\mathrm{x}}{\left(\mathrm{a}^{\mathrm{2}} \mathrm{cosx}+\mathrm{b}^{\mathrm{2}} \mathrm{sinx}\right)}\mathrm{dx} \\ $$

Question Number 103454 Answers: 1 Comments: 0

∫ (x^2 +2x^4 +3x^6 )(√(1+x^2 +x^4 )) dx

$$\int\:\left({x}^{\mathrm{2}} +\mathrm{2}{x}^{\mathrm{4}} +\mathrm{3}{x}^{\mathrm{6}} \right)\sqrt{\mathrm{1}+{x}^{\mathrm{2}} +{x}^{\mathrm{4}} }\:{dx}\: \\ $$

Question Number 103397 Answers: 1 Comments: 0

I(n)=∫_0 ^∞ ((ln x)/(cosh^n x ))dx is there a simpler way to calculat those values

$${I}\left({n}\right)=\int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\:{x}}{\mathrm{cosh}^{{n}} {x}\:}{dx} \\ $$$${is}\:{there}\:{a}\:{simpler}\:{way}\:{to} \\ $$$${calculat}\:{those}\:{values} \\ $$

Question Number 103343 Answers: 1 Comments: 0

∫_0 ^1 x^(−x) dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{−{x}} {dx} \\ $$

Question Number 103312 Answers: 4 Comments: 0

∫_0 ^∞ (1/((1+x^2 )^6 )) dx ?

$$\underset{\mathrm{0}} {\overset{\infty} {\int}}\:\frac{\mathrm{1}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{\mathrm{6}} }\:{dx}\:? \\ $$

Question Number 103241 Answers: 0 Comments: 0

∫x^(−x) dx

$$\int{x}^{−{x}} {dx} \\ $$

Question Number 103220 Answers: 1 Comments: 0

∫_0 ^∞ (x^3 /(e^x +1))dx

$$\int_{\mathrm{0}} ^{\infty} \frac{{x}^{\mathrm{3}} }{{e}^{{x}} +\mathrm{1}}{dx} \\ $$

Question Number 103198 Answers: 4 Comments: 1

∫_0 ^1 sin(logx)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {sin}\left({logx}\right){dx} \\ $$

Question Number 103196 Answers: 1 Comments: 0

∫_0 ^1 ((((1/2)−x) ln(1−x) dx)/(x^2 −x+1)) ?

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\left(\frac{\mathrm{1}}{\mathrm{2}}−{x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}\right)\:{dx}}{{x}^{\mathrm{2}} −{x}+\mathrm{1}}\:? \\ $$

Question Number 103154 Answers: 2 Comments: 0

∫_0 ^1 logxlog(1−x)dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} {logxlog}\left(\mathrm{1}−{x}\right){dx} \\ $$

Question Number 103089 Answers: 1 Comments: 0

solve: (sin^2 (x)−y)dx−tan(x)dy=0

$${solve}: \\ $$$$\left({sin}^{\mathrm{2}} \left({x}\right)−{y}\right){dx}−{tan}\left({x}\right){dy}=\mathrm{0} \\ $$

Question Number 103080 Answers: 0 Comments: 0

calculate A_n =∫_0 ^∞ ((cos(nx))/((1+x^2 )^n ))dx with n integr natural

$${calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\frac{{cos}\left({nx}\right)}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)^{{n}} }{dx} \\ $$$${with}\:{n}\:{integr}\:{natural} \\ $$

Question Number 103078 Answers: 0 Comments: 0

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

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

Question Number 103077 Answers: 0 Comments: 0

find ∫_0 ^∞ ((x^2 cosx)/((x^2 +x+2)^2 ))dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\mathrm{2}} {cosx}}{\left({x}^{\mathrm{2}} +{x}+\mathrm{2}\right)^{\mathrm{2}} }{dx} \\ $$

Question Number 103040 Answers: 5 Comments: 0

given 5x+12y = 60 min value of (√(x^2 +y^2 ))

$${given}\:\mathrm{5}{x}+\mathrm{12}{y}\:=\:\mathrm{60} \\ $$$${min}\:{value}\:{of}\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} } \\ $$

Question Number 102987 Answers: 2 Comments: 0

lim_(x→0) ((x^2 sin (x^(−4) ))/x) ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}^{\mathrm{2}} \:\mathrm{sin}\:\left({x}^{−\mathrm{4}} \right)}{{x}}\:?\: \\ $$

Question Number 102926 Answers: 3 Comments: 0

∫ (dθ/(2sin^2 θ−cos^2 θ)) ?

$$\:\int\:\frac{{d}\theta}{\mathrm{2}{sin}^{\mathrm{2}} \theta−{cos}^{\mathrm{2}} \theta}\:\:? \\ $$

Question Number 102922 Answers: 1 Comments: 1

∫ (dx/(x^3 +3x−5)) ?

$$\int\:\frac{{dx}}{{x}^{\mathrm{3}} +\mathrm{3}{x}−\mathrm{5}}\:\:? \\ $$

Question Number 103071 Answers: 0 Comments: 0

∫(e^x /((1+x^2 )^2 ))∙(x^3 −x^2 +x+1)dx

$$\int\frac{\mathrm{e}^{\mathrm{x}} }{\left(\mathrm{1}+\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{2}} }\centerdot\left(\mathrm{x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}\right)\mathrm{dx} \\ $$

Question Number 102911 Answers: 2 Comments: 1

∫ ((sin(x))/x) dx

$$\int\:\frac{\mathrm{sin}\left(\mathrm{x}\right)}{\mathrm{x}}\:\:\mathrm{dx} \\ $$

Question Number 102905 Answers: 1 Comments: 0

I=2∫_0 ^(1/(√2)) ((sin^(−1) (x))/x) dx −∫_0 ^1 ((tan^(−1) (x))/x)dx

$${I}=\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}} {\int}}\:\frac{\mathrm{sin}^{−\mathrm{1}} \left({x}\right)}{{x}}\:{dx}\:−\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{{x}}{dx} \\ $$

Question Number 102790 Answers: 1 Comments: 0

I=2∫_0 ^(1/(√2)) ((sin^(−1) (x))/x) dx −∫_0 ^1 ((tan^(−1) (x))/x) dx

$${I}=\mathrm{2}\underset{\mathrm{0}} {\overset{\frac{\mathrm{1}}{\sqrt{\mathrm{2}}}} {\int}}\:\frac{\mathrm{sin}^{−\mathrm{1}} \left({x}\right)}{{x}}\:{dx}\:−\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{\mathrm{tan}^{−\mathrm{1}} \left({x}\right)}{{x}}\:{dx}\: \\ $$

Question Number 102894 Answers: 3 Comments: 0

∫ (√(x+(√(x+(√(x+(√(x+(√(x+...)))))))))) dx

$$\int\:\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+...}}}}}\:\mathrm{dx}\: \\ $$

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