Question and Answers Forum

IntegrationQuestion and Answers: Page 19

Question Number 208129 Answers: 2 Comments: 0

x^(303) + x^(73) + 1 ∫_1 ^(379) ^(−1) (x)

$$\:\:\:\: \mathrm{x}^{\mathrm{303}} \:+\: \mathrm{x}^{\mathrm{73}} \:+\:\mathrm{1}\: \\ $$$$\:\:\:\underset{\mathrm{1}} {\overset{\mathrm{379}} {\int}} ^{−\mathrm{1}} \left(\mathrm{x}\right)\: \\ $$

Question Number 208128 Answers: 0 Comments: 0

Question Number 208062 Answers: 1 Comments: 0

find ∫_4 ^∞ (dx/((x+1)^3 (x−3)^5 ))

$${find}\:\:\int_{\mathrm{4}} ^{\infty} \:\:\:\:\frac{{dx}}{\left({x}+\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{3}\right)^{\mathrm{5}} } \\ $$

Question Number 208052 Answers: 0 Comments: 3

Sketch the curve y = x^3 . (a) Find the equation of the tangent to the curve at A(1,1). (b) Find the coordinates of point B, where the tangent meets the curve again. (c) Calculate the area between the tangent B and the arc AB of the curve.

$${Sketch}\:{the}\:{curve}\:{y}\:=\:{x}^{\mathrm{3}} . \\ $$$$\left({a}\right)\:{Find}\:{the}\:{equation}\:{of}\:{the}\:{tangent} \\ $$$${to}\:{the}\:{curve}\:{at}\:{A}\left(\mathrm{1},\mathrm{1}\right). \\ $$$$\left({b}\right)\:{Find}\:{the}\:{coordinates}\:{of}\:{point}\:{B}, \\ $$$${where}\:{the}\:{tangent}\:{meets}\:{the}\:{curve}\:{again}. \\ $$$$\left({c}\right)\:{Calculate}\:{the}\:{area}\:{between}\:{the} \\ $$$${tangent}\:{B}\:{and}\:{the}\:{arc}\:{AB}\:{of}\:{the}\:{curve}. \\ $$

Question Number 207949 Answers: 2 Comments: 0

Find the value of : 𝛀 = ∫_0 ^( (𝛑/2)) (( dx)/(sin^6 x + cos^6 x)) = ? −−−−−−−−−

$$ \\ $$$$\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:{Find}\:{the}\:{value}\:{of}\:: \\ $$$$ \\ $$$$\:\:\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{2}}} \:\frac{\:\boldsymbol{{dx}}}{\boldsymbol{{sin}}^{\mathrm{6}} \boldsymbol{{x}}\:+\:\boldsymbol{{cos}}^{\mathrm{6}} \boldsymbol{{x}}}\:=\:?\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$

Question Number 207938 Answers: 1 Comments: 0

what is the area bounded by the curve y=x(x−2)(x−5) and the x axis?

$${what}\:{is}\:{the}\:{area}\:{bounded}\:{by}\:{the}\:{curve} \\ $$$${y}={x}\left({x}−\mathrm{2}\right)\left({x}−\mathrm{5}\right)\:{and}\:{the}\:{x}\:{axis}? \\ $$$$ \\ $$

Question Number 207924 Answers: 0 Comments: 0

∫f(x)g(x)dx=Σ_(n=0) ^∞ (−1)^n lim_(h→0) (1/h^n ) Σ_(i=o) ^n [ (−1)^i (((n!)/(i!(n−i)!)))f(x+(n−i)h)] (1/(n!))∫_a ^x (x−t)^n g(t)dt prove that right its a relation that i have derrived

$$\int{f}\left({x}\right){g}\left({x}\right){dx}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\left(−\mathrm{1}\right)^{{n}} \:\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{{h}^{{n}} }\:\underset{{i}={o}} {\overset{{n}} {\sum}}\left[\:\left(−\mathrm{1}\right)^{{i}} \left(\frac{{n}!}{{i}!\left({n}−{i}\right)!}\right){f}\left({x}+\left({n}−{i}\right){h}\right)\right]\:\frac{\mathrm{1}}{{n}!}\underset{{a}} {\overset{{x}} {\int}}\left({x}−{t}\right)^{{n}} {g}\left({t}\right){dt}\: \\ $$$${prove}\:{that}\:{right} \\ $$$${its}\:{a}\:{relation}\:{that}\:{i}\:{have}\:{derrived} \\ $$

Question Number 207906 Answers: 1 Comments: 0

help ∫_1 ^( ∞) x^(−ln(x)) dx

$${help} \\ $$$$\int_{\mathrm{1}} ^{\:\infty} {x}^{−{ln}\left({x}\right)} {dx} \\ $$$$ \\ $$

Question Number 207878 Answers: 2 Comments: 0

Question Number 207857 Answers: 1 Comments: 0

∫xtan^(−1) xdx

$$\int{x}\mathrm{tan}^{−\mathrm{1}} {xdx} \\ $$

Question Number 207789 Answers: 1 Comments: 0

∀r∈R: H_r =∫_0 ^1 ((t^r −1)/(t−1))dt H_(r+2) −H_r =1 r=?

$$\forall{r}\in\mathbb{R}:\:{H}_{{r}} =\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{t}^{{r}} −\mathrm{1}}{{t}−\mathrm{1}}{dt} \\ $$$${H}_{{r}+\mathrm{2}} −{H}_{{r}} =\mathrm{1} \\ $$$${r}=? \\ $$

Question Number 207753 Answers: 1 Comments: 0

Question Number 207707 Answers: 0 Comments: 0

∫_0 ^(+∞) ((sin^2 (x))/(sin^2 (x)+(xcos (x)+sin (x))^2 ))d(x)

$$\underset{\mathrm{0}} {\overset{+\infty} {\int}}\frac{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)}{\mathrm{sin}\:^{\mathrm{2}} \left({x}\right)+\left({x}\mathrm{cos}\:\left({x}\right)+\mathrm{sin}\:\left({x}\right)\right)^{\mathrm{2}} }{d}\left({x}\right) \\ $$

Question Number 207652 Answers: 1 Comments: 0

∫_0 ^1 log(1+x^3 )dx = ?and ∫_0 ^1 log (1+x^4 )dx = ? and if possible then find the value of p p = ∫_0 ^1 log(1+x^n )dx = ? n∈N

$$\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx}\:\:=\:?{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx}\:=\:? \\ $$$$\:\:\mathrm{and}\:\mathrm{if}\:\mathrm{possible}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{p} \\ $$$$\:\mathrm{p}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{{n}} \right){dx}\:=\:?\:\:\:\:\:\:{n}\in\mathbb{N} \\ $$

Question Number 207582 Answers: 2 Comments: 0

∫_0 ^π ln(sinx)dx=−πln2 ∫_0 ^1 lnΓ(x)dx = ln(2π)

$$\int_{\mathrm{0}} ^{\pi} \:{ln}\left({sinx}\right){dx}=−\pi{ln}\mathrm{2} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\Gamma\left({x}\right){dx}\:=\:{ln}\left(\mathrm{2}\pi\right) \\ $$

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