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IntegrationQuestion and Answers: Page 20

Question Number 207565 Answers: 2 Comments: 0

find ∫_0 ^(π/2) (x^2 /(tan^2 x))dx

$${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{{tan}^{\mathrm{2}} {x}}{dx} \\ $$

Question Number 207424 Answers: 0 Comments: 0

f_n (x):=∫e^((2x)/3) ((cos(x))/( (cos(x)+sin(x))^(n/3) ))dx=...? for n=1, i found f_1 (x)=(3/4)e^((2x)/3) (cos(x)+sin(x))^(2/3) + C is there any ideas for a general case or the case n=2?

$${f}_{{n}} \left({x}\right):=\int{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \frac{{cos}\left({x}\right)}{\:\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{{n}}{\mathrm{3}}} }{dx}=...? \\ $$$${for}\:{n}=\mathrm{1},\:{i}\:{found}\: \\ $$$$\:\:\:\:\:\:{f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}}{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\:{C} \\ $$$${is}\:{there}\:{any}\:{ideas}\:{for}\:{a}\:{general}\:{case}\:{or} \\ $$$${the}\:{case}\:{n}=\mathrm{2}? \\ $$

Question Number 207383 Answers: 0 Comments: 1

∫((ln(x^2 +sin(sin(e^x ))))/( (√(x+tan(ln(x))))))dx

$$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$

Question Number 207382 Answers: 1 Comments: 0

Question Number 207352 Answers: 1 Comments: 3

calculate: ∫_(Π/4) ^(Π/2) ⌊cot(x)⌋ dx

$${calculate}: \\ $$$$\:\int_{\frac{\Pi}{\mathrm{4}}} ^{\frac{\Pi}{\mathrm{2}}} \lfloor{cot}\left({x}\right)\rfloor\:{dx} \\ $$

Question Number 207354 Answers: 0 Comments: 4

Question Number 207359 Answers: 1 Comments: 0

∫((ln(x^2 +sin(sin(e^x ))))/( (√(x+tan(ln(x))))))dx

$$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$

Question Number 207099 Answers: 1 Comments: 0

Question Number 207054 Answers: 1 Comments: 0

Ω_α =∫_0 ^1 x^α (√(−xln x)) dx=?

$$\Omega_{\alpha} =\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{x}^{\alpha} \sqrt{−{x}\mathrm{ln}\:{x}}\:{dx}=? \\ $$

Question Number 206962 Answers: 2 Comments: 0

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

$$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\sqrt{\mathrm{1}−{x}}}{\:\sqrt{\mathrm{1}−\sqrt{\mathrm{1}−{x}}}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}−{x}}}}{dx}=? \\ $$

Question Number 206892 Answers: 1 Comments: 0

find ∫_0 ^1 (√(1+(√(1+x^2 ))))dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 206890 Answers: 0 Comments: 1

can some one find the exact value of Σ_(n=0) ^∞ (1/((n!)^2 ))

$${can}\:{some}\:{one}\:{find}\:{the}\:{exact}\:{value}\:{of} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}!\right)^{\mathrm{2}} } \\ $$

Question Number 206858 Answers: 2 Comments: 0

prove that H_n =∫_0 ^1 ((t^n −1)/(t−1))dt

$$\mathrm{prove}\:\mathrm{that} \\ $$$${H}_{{n}} =\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{t}^{{n}} −\mathrm{1}}{{t}−\mathrm{1}}{dt} \\ $$

Question Number 206830 Answers: 0 Comments: 0

c = (√((∫_a_0 ^a_1 (√(1+[f′(x)]^2 ))dx)^2 +(∫_b_0 ^b_1 (√(1+[f′(x)]^2 ))dx)^2 )) c = (√(L_1 ^2 +L_2 ^2 ))

$${c}\:=\:\sqrt{\left(\int_{{a}_{\mathrm{0}} } ^{{a}_{\mathrm{1}} } \sqrt{\mathrm{1}+\left[{f}'\left({x}\right)\right]^{\mathrm{2}} }{dx}\right)^{\mathrm{2}} +\left(\int_{{b}_{\mathrm{0}} } ^{{b}_{\mathrm{1}} } \sqrt{\mathrm{1}+\left[{f}'\left({x}\right)\right]^{\mathrm{2}} }{dx}\right)^{\mathrm{2}} } \\ $$$${c}\:=\:\sqrt{{L}_{\mathrm{1}} ^{\mathrm{2}} +{L}_{\mathrm{2}} ^{\mathrm{2}} } \\ $$

Question Number 206829 Answers: 0 Comments: 1

∮(x/(x+2))dx^2 is wrong?

$$\:\:\:\:\:\oint\frac{{x}}{{x}+\mathrm{2}}{dx}^{\mathrm{2}} \:\:\:\:{is}\:{wrong}? \\ $$

Question Number 206789 Answers: 1 Comments: 0

Question Number 206754 Answers: 1 Comments: 0

find ∫_0 ^1 (√(1−(√x)))ln^2 (x)dx

$${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−\sqrt{{x}}}{ln}^{\mathrm{2}} \left({x}\right){dx} \\ $$

Question Number 206721 Answers: 4 Comments: 0

∫((xdx)/(x+4))=? please

$$\int\frac{{xdx}}{{x}+\mathrm{4}}=?\:\:\:\:\:\:\:{please} \\ $$

Question Number 206791 Answers: 0 Comments: 3

∫_0 ^1 (√(1−(√x))).ln^2 x dx

$$\int_{\mathrm{0}} ^{\mathrm{1}} \sqrt{\mathrm{1}−\sqrt{\mathrm{x}}}.\mathrm{ln}^{\mathrm{2}} \mathrm{x}\:\mathrm{dx} \\ $$

Question Number 206645 Answers: 1 Comments: 5

Let f(x)=x(x−10) and let A be the region enclosed within the following points (2,7),(8,7),(2,4),(8,4) what is the average arc length of a∙f(x) inside A,a∈R^−

$$\mathrm{Let}\:{f}\left({x}\right)={x}\left({x}−\mathrm{10}\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:\mathrm{A}\:\mathrm{be}\:\mathrm{the}\:\mathrm{region}\:\mathrm{enclosed}\:\mathrm{within} \\ $$$$\mathrm{the}\:\mathrm{following}\:\mathrm{points} \\ $$$$\left(\mathrm{2},\mathrm{7}\right),\left(\mathrm{8},\mathrm{7}\right),\left(\mathrm{2},\mathrm{4}\right),\left(\mathrm{8},\mathrm{4}\right) \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{average}\:\mathrm{arc}\:\mathrm{length}\:\mathrm{of}\:\mathrm{a}\centerdot{f}\left({x}\right) \\ $$$$\mathrm{inside}\:\mathrm{A},{a}\in\mathbb{R}^{−} \\ $$

Question Number 206642 Answers: 0 Comments: 0

Question Number 206639 Answers: 0 Comments: 1

Question Number 206592 Answers: 2 Comments: 0

Question Number 206579 Answers: 1 Comments: 0

Evaluate : ∫_0 ^1 ((ln (1+x^2 ))/(1+x))d(x).

$$\mathrm{Evaluate}\::\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{\mathrm{ln}\:\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}{d}\left({x}\right). \\ $$

Question Number 206568 Answers: 1 Comments: 0

Question Number 206558 Answers: 1 Comments: 0

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