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IntegrationQuestion and Answers: Page 194

Question Number 79500 Answers: 1 Comments: 0

∫(cot^2 x+cot^4 x)dx

$$\int\left(\mathrm{cot}\:^{\mathrm{2}} {x}+\mathrm{cot}\:^{\mathrm{4}} {x}\right){dx} \\ $$

Question Number 79485 Answers: 1 Comments: 0

∫(tan^2 x+tan^4 x)dx

$$\int\left(\mathrm{tan}\:^{\mathrm{2}} {x}+\mathrm{tan}\:^{\mathrm{4}} {x}\right){dx} \\ $$

Question Number 79413 Answers: 0 Comments: 1

Question Number 79373 Answers: 0 Comments: 0

find ∫_0 ^∞ e^(−(x^3 +(1/x^3 ))) dx

$${find}\:\int_{\mathrm{0}} ^{\infty} \:{e}^{−\left({x}^{\mathrm{3}} \:+\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)} {dx} \\ $$

Question Number 79352 Answers: 1 Comments: 2

∫(dx/(1−(√(cos(x)))))

$$\int\frac{{dx}}{\mathrm{1}−\sqrt{{cos}\left({x}\right)}} \\ $$

Question Number 79325 Answers: 0 Comments: 2

Convergence of : 1) I=∫_1 ^( ∞) ((e^(−t/5) ∣sin(lnt)∣)/((t−1)^(3/2) ))dt 2) I=∫_1 ^∞ ((√(lnx))/((x−1)(√x)))dx

$$\:\boldsymbol{{Convergence}}\:\:\boldsymbol{{of}}\:: \\ $$$$\left.\:\:\mathrm{1}\right)\:\:\:\boldsymbol{{I}}=\int_{\mathrm{1}} ^{\:\infty} \frac{\boldsymbol{{e}}^{−\boldsymbol{{t}}/\mathrm{5}} \mid\boldsymbol{{sin}}\left(\boldsymbol{{lnt}}\right)\mid}{\left(\boldsymbol{{t}}−\mathrm{1}\right)^{\mathrm{3}/\mathrm{2}} }\boldsymbol{{dt}} \\ $$$$\left.\:\:\mathrm{2}\right)\:\:\:\boldsymbol{{I}}=\int_{\mathrm{1}} ^{\infty} \frac{\sqrt{\boldsymbol{{lnx}}}}{\left(\boldsymbol{{x}}−\mathrm{1}\right)\sqrt{\boldsymbol{{x}}}}\boldsymbol{{dx}} \\ $$

Question Number 79222 Answers: 0 Comments: 3

∫_0 ^1 (x^n /(Σ_(k=0) ^(n−1) x^k ))dx=?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\sum}}{x}^{{k}} }{dx}=? \\ $$

Question Number 79187 Answers: 0 Comments: 0

∫_0 ^π ((cos (nx)−cos (nα))/(cos (x)−cos (α))) dx

$$\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\frac{\mathrm{cos}\:\left({nx}\right)−\mathrm{cos}\:\left({n}\alpha\right)}{\mathrm{cos}\:\left({x}\right)−\mathrm{cos}\:\left(\alpha\right)}\:{dx} \\ $$

Question Number 79186 Answers: 1 Comments: 0

∫_(−1) ^1 ((cos (x))/(1+e^(1/x) )) dx ?

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

Question Number 79128 Answers: 1 Comments: 4

Find out ∫_0 ^1 ln(1−t+t^2 )dt Then deduce the value of A=Σ_(n=1) ^∞ (1/(n(n+1) (((2n+1)),(n) )))

$${Find}\:{out}\:\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\left(\mathrm{1}−{t}+{t}^{\mathrm{2}} \right){dt} \\ $$$${Then}\:{deduce}\:{the}\:{value}\:{of}\:\:\:{A}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)\begin{pmatrix}{\mathrm{2}{n}+\mathrm{1}}\\{{n}}\end{pmatrix}} \\ $$

Question Number 79100 Answers: 0 Comments: 1

calculate ∫_0 ^∞ (((−1)^x^2 )/((x^2 −x+1)^3 ))dx

$${calculate}\:\:\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} −{x}+\mathrm{1}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 79096 Answers: 1 Comments: 1

calculate ∫_0 ^∞ ((ln(x))/((1+x)^3 ))dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{{ln}\left({x}\right)}{\left(\mathrm{1}+{x}\right)^{\mathrm{3}} }{dx} \\ $$

Question Number 79095 Answers: 0 Comments: 1

find A_n =∫_0 ^∞ ((sin(x)sin(2x)....sin(nx))/x^n )dx with n≥2 integr

$${find}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({x}\right){sin}\left(\mathrm{2}{x}\right)....{sin}\left({nx}\right)}{{x}^{{n}} }{dx}\:\:{with}\:{n}\geqslant\mathrm{2}\:{integr} \\ $$

Question Number 79094 Answers: 1 Comments: 0

find I_(a,b) =∫_0 ^∞ ((sin(ax)sin(bx))/x^2 )dx witha>0 and b>0

$${find}\:{I}_{{a},{b}} \:=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sin}\left({ax}\right){sin}\left({bx}\right)}{{x}^{\mathrm{2}} }{dx}\:\:\:{witha}>\mathrm{0}\:{and}\:{b}>\mathrm{0} \\ $$

Question Number 79093 Answers: 0 Comments: 0

find f(λ) =∫_0 ^∞ e^(−λx^2 ) ch(x^2 +λ)dx with λ>0

$${find}\:\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−\lambda{x}^{\mathrm{2}} } {ch}\left({x}^{\mathrm{2}} \:+\lambda\right){dx}\:\:{with}\:\lambda>\mathrm{0} \\ $$

Question Number 79092 Answers: 0 Comments: 0

find ∫_(−∞) ^(+∞) ((e^(−x^2 ) arctan(x^2 +1))/(x^2 +1))dx

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

Question Number 79091 Answers: 0 Comments: 0

calculate ∫_0 ^∞ ((e^(−x^2 ) arctan(x))/x)dx

$${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{e}^{−{x}^{\mathrm{2}} } \:{arctan}\left({x}\right)}{{x}}{dx} \\ $$

Question Number 79086 Answers: 0 Comments: 2

if:∫cos(f(x))dx=g(x) ∫sin(f(x))dx=? (use g(x))

$$\mathrm{if}:\int\mathrm{cos}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=\mathrm{g}\left(\mathrm{x}\right) \\ $$$$\int\mathrm{sin}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{dx}=?\:\left(\mathrm{use}\:\mathrm{g}\left(\mathrm{x}\right)\right) \\ $$

Question Number 79059 Answers: 0 Comments: 4

∫ cos^2 (x)sin^4 (x) dx ?

$$ \\ $$$$ \\ $$$$\int\:\mathrm{cos}\:^{\mathrm{2}} \left(\mathrm{x}\right)\mathrm{sin}\:^{\mathrm{4}} \left(\mathrm{x}\right)\:\mathrm{dx}\:? \\ $$

Question Number 78998 Answers: 0 Comments: 0

The value of ∫_0 ^(pi) sin2xdx+2∫_0 ^(pi/2) cos2xdx

$${The}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{pi}} \mathrm{sin2xdx}+\mathrm{2}\int_{\mathrm{0}} ^{{pi}/\mathrm{2}} {cos}\mathrm{2}{xdx} \\ $$

Question Number 78897 Answers: 1 Comments: 11

Question Number 78853 Answers: 4 Comments: 0

∫_0 ^(π/2) ∫_0 ^(π/2) ((sin(x)+sin(y))/(x+y))dxdy?

$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}\left(\mathrm{x}\right)+\mathrm{sin}\left(\mathrm{y}\right)}{\mathrm{x}+\mathrm{y}}\mathrm{dxdy}? \\ $$

Question Number 78829 Answers: 0 Comments: 8

given f(x)=f(x+4) ∀x∈R and ∫_5 ^7 f(x)dx=p . what is ∫_2 ^(10) f(x)dx?

$$\mathrm{given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{4}\right)\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\mathrm{and}\:\underset{\mathrm{5}} {\overset{\mathrm{7}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{p}\:.\:\mathrm{what}\:\mathrm{is}\: \\ $$$$\underset{\mathrm{2}} {\overset{\mathrm{10}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}? \\ $$$$ \\ $$

Question Number 78797 Answers: 1 Comments: 0

Show that: ∫_( 0) ^( ∞) (x^3 /(e^x − 1)) dx = (π^4 /(15))

$$\mathrm{Show}\:\mathrm{that}:\:\:\:\:\int_{\:\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{e}^{\mathrm{x}} \:−\:\mathrm{1}}\:\mathrm{dx}\:\:\:\:=\:\:\:\frac{\pi^{\mathrm{4}} }{\mathrm{15}} \\ $$

Question Number 78766 Answers: 1 Comments: 0

∫2 e^(1/(2(x−2)^2 )) dx

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

Question Number 78717 Answers: 0 Comments: 2

given ∫ f(x) dx = (1/(2 ((g(x)))^(1/(3 )) )) . g′(1)= g(1) = 8 ⇒f(1)=?

$$\mathrm{given}\: \\ $$$$\int\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}\:\sqrt[{\mathrm{3}\:}]{\mathrm{g}\left(\mathrm{x}\right)}}\:.\: \\ $$$$\mathrm{g}'\left(\mathrm{1}\right)=\:\mathrm{g}\left(\mathrm{1}\right)\:=\:\mathrm{8}\:\Rightarrow\mathrm{f}\left(\mathrm{1}\right)=? \\ $$$$ \\ $$

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