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IntegrationQuestion and Answers: Page 77

Question Number 150871 Answers: 1 Comments: 0

Find all the real solutions of cos x+cos^5 x+cos 7x=3

$$\:\mathrm{Find}\:\mathrm{all}\:\mathrm{the}\:\mathrm{real}\:\mathrm{solutions} \\ $$$$\mathrm{of}\:\mathrm{cos}\:\mathrm{x}+\mathrm{cos}\:^{\mathrm{5}} \mathrm{x}+\mathrm{cos}\:\mathrm{7x}=\mathrm{3} \\ $$

Question Number 148720 Answers: 4 Comments: 0

Find the Talor series of ((ln(1−x))/((1−x)^2 )) at x=0.

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{Talor}\:\mathrm{series}\:\mathrm{of}\:\frac{\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}{\left(\mathrm{1}−\mathrm{x}\right)^{\mathrm{2}} }\:\mathrm{at}\:\mathrm{x}=\mathrm{0}. \\ $$

Question Number 148570 Answers: 2 Comments: 0

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

$$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\frac{\mathrm{logx}}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}}\mathrm{dx} \\ $$

Question Number 148567 Answers: 1 Comments: 0

find ∫_0 ^∞ ((x^2 logx)/((x^2 +1)^3 ))dx

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

Question Number 148566 Answers: 0 Comments: 0

calculate U_n =∫∫_([(1/n),n[) ((cos(x^2 +y^2 ))/(x^2 +y^2 ))dxdy and determine lim_(n→+∞) U_n nature of Σ U_n ?

$$\mathrm{calculate}\:\:\mathrm{U}_{\mathrm{n}} =\int\int_{\left[\frac{\mathrm{1}}{\mathrm{n}},\mathrm{n}\left[\right.\right.} \:\:\:\frac{\mathrm{cos}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} \right)}{\mathrm{x}^{\mathrm{2}} \:+\mathrm{y}^{\mathrm{2}} }\mathrm{dxdy} \\ $$$$\mathrm{and}\:\mathrm{determine}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \mathrm{U}_{\mathrm{n}} \\ $$$$\mathrm{nature}\:\mathrm{of}\:\Sigma\:\mathrm{U}_{\mathrm{n}} ? \\ $$

Question Number 148565 Answers: 2 Comments: 0

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

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

Question Number 148467 Answers: 2 Comments: 0

Question Number 148441 Answers: 0 Comments: 0

Σ_(k=1) ^∞ Σ_(m=1) ^n ((n(m−1))/((nk+m−1)(nk+m)))=?

$$\underset{\mathrm{k}=\mathrm{1}} {\overset{\infty} {\sum}}\underset{\mathrm{m}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{n}\left(\mathrm{m}−\mathrm{1}\right)}{\left(\mathrm{nk}+\mathrm{m}−\mathrm{1}\right)\left(\mathrm{nk}+\mathrm{m}\right)}=? \\ $$

Question Number 148408 Answers: 2 Comments: 0

∫x^5 e^x^2 dx Help please!

$$\int{x}^{\mathrm{5}} {e}^{{x}^{\mathrm{2}} } {dx} \\ $$$${Help}\:{please}! \\ $$

Question Number 148376 Answers: 1 Comments: 0

∫_1 ^∞ x^i lnxdx i^2 =−1

$$\int_{\mathrm{1}} ^{\infty} {x}^{{i}} {lnxdx}\:\:\:\:\:\:{i}^{\mathrm{2}} =−\mathrm{1} \\ $$

Question Number 148375 Answers: 0 Comments: 0

∫_1 ^∞ sin(x+lnx)dx

$$\int_{\mathrm{1}} ^{\infty} {sin}\left({x}+{lnx}\right){dx} \\ $$

Question Number 148231 Answers: 0 Comments: 1

∫_0 ^1 x^dx =?

$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{dx}} =? \\ $$

Question Number 148128 Answers: 1 Comments: 0

∫ln(sin(x))dx

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

Question Number 148026 Answers: 1 Comments: 0

A :=∫_(−1) ^( 0) e^( x +(1/x)) & aA+e^b = ∫_0 ^( ∞) ((1/e))^( x+(1/x)) than : a+ b =? a , b ∈ Z

$$ \\ $$$$\mathrm{A}\::=\int_{−\mathrm{1}} ^{\:\mathrm{0}} {e}^{\:{x}\:+\frac{\mathrm{1}}{{x}}} \:\:\&\:{a}\mathrm{A}+{e}^{{b}} =\:\int_{\mathrm{0}} ^{\:\infty} \left(\frac{\mathrm{1}}{{e}}\right)^{\:{x}+\frac{\mathrm{1}}{{x}}} \\ $$$$\:\:{than}\::\:\:{a}+\:{b}\:=?\:\:\:\:\:{a}\:,\:{b}\:\in\:\mathbb{Z} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 148020 Answers: 0 Comments: 0

Σ_(k=2) ^∞ (−1)^k ∙((lnk)/k)=γln2−(1/2)ln^2 2

$$\underset{\mathrm{k}=\mathrm{2}} {\overset{\infty} {\sum}}\left(−\mathrm{1}\right)^{\mathrm{k}} \centerdot\frac{\mathrm{lnk}}{\mathrm{k}}=\gamma\mathrm{ln2}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{ln}^{\mathrm{2}} \mathrm{2} \\ $$

Question Number 148000 Answers: 2 Comments: 0

∫_(−∞) ^(+∞) (dx/((x^2 +k^2 )^(3/2) ))

$$\underset{−\infty} {\overset{+\infty} {\int}}\frac{\boldsymbol{{dx}}}{\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{k}}^{\mathrm{2}} \right)^{\frac{\mathrm{3}}{\mathrm{2}}} } \\ $$

Question Number 147884 Answers: 0 Comments: 0

∫_0 ^1 e^(−x) x^a dx a>0

$$\int_{\mathrm{0}} ^{\mathrm{1}} {e}^{−{x}} {x}^{{a}} {dx}\:\:{a}>\mathrm{0} \\ $$$$ \\ $$

Question Number 147803 Answers: 0 Comments: 7

∫(((√(2−x^2 ))+(√(2+x^2 )))/( (√(4−x^4 ))))dx

$$\int\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{4}−{x}^{\mathrm{4}} }}{dx} \\ $$

Question Number 147799 Answers: 1 Comments: 0

∫(((√(2−x^2 ))+(√(2+x^2 )))/( (√(4−x^2 ))))dx

$$\int\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }}{dx} \\ $$

Question Number 147749 Answers: 1 Comments: 0

∫(((√(2−x^2 ))+(√(2+x^2 )))/( (√(4−x^4 ))))dx

$$\int\frac{\sqrt{\mathrm{2}−{x}^{\mathrm{2}} }+\sqrt{\mathrm{2}+{x}^{\mathrm{2}} }}{\:\sqrt{\mathrm{4}−{x}^{\mathrm{4}} }}{dx} \\ $$

Question Number 147747 Answers: 0 Comments: 0

Question Number 147746 Answers: 1 Comments: 0

Question Number 147683 Answers: 1 Comments: 0

let F(x)=(1/((x+1)^5 (2x−3)^4 )) 1) find ∫ F(x)dx 2)en deduire la decomposition de F en element simples

$$\mathrm{let}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{5}} \left(\mathrm{2x}−\mathrm{3}\right)^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:\mathrm{find}\:\int\:\mathrm{F}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\left.\mathrm{2}\right)\mathrm{en}\:\mathrm{deduire}\:\mathrm{la}\:\mathrm{decomposition}\:\mathrm{de}\:\mathrm{F}\:\mathrm{en}\:\mathrm{element}\:\mathrm{simples} \\ $$

Question Number 147682 Answers: 0 Comments: 0

decompose F(x)=(1/((x^n −1)(x^2 +x+1))) dans C(x) puis dans R(x)

$$\mathrm{decompose}\:\mathrm{F}\left(\mathrm{x}\right)=\frac{\mathrm{1}}{\left(\mathrm{x}^{\mathrm{n}} −\mathrm{1}\right)\left(\mathrm{x}^{\mathrm{2}} \:+\mathrm{x}+\mathrm{1}\right)}\:\mathrm{dans}\:\mathrm{C}\left(\mathrm{x}\right)\:\mathrm{puis}\:\mathrm{dans}\:\mathrm{R}\left(\mathrm{x}\right) \\ $$

Question Number 147680 Answers: 0 Comments: 2

find by residus ∫_0 ^∞ ((cos(2x))/((x^2 −x+1)^3 ))dx

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

Question Number 147670 Answers: 1 Comments: 0

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