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IntegrationQuestion and Answers: Page 23

Question Number 203747    Answers: 2   Comments: 0

Question Number 203679    Answers: 0   Comments: 0

Question Number 203564    Answers: 1   Comments: 0

∫_0 ^( ∞) ((sin^( 3) (x))/x^( 2) ) dx= ?

$$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \frac{{sin}^{\:\mathrm{3}} \left({x}\right)}{{x}^{\:\mathrm{2}} }\:{dx}=\:?\:\:\:\:\: \\ $$

Question Number 203385    Answers: 2   Comments: 0

Question Number 203349    Answers: 1   Comments: 0

calculate ∫∫_([0,a]^2 ) e^(−x^2 −y^2 ) dxdy can you find ∫_0 ^a e^(−x^2 ) dx ? a>0

$${calculate}\:\int\int_{\left[\mathrm{0},{a}\right]^{\mathrm{2}} } \:{e}^{−{x}^{\mathrm{2}} −{y}^{\mathrm{2}} } {dxdy} \\ $$$${can}\:{you}\:{find}\:\int_{\mathrm{0}} ^{{a}} {e}^{−{x}^{\mathrm{2}} } {dx}\:\:\:\:? \\ $$$${a}>\mathrm{0} \\ $$

Question Number 203186    Answers: 1   Comments: 0

f(x)={_(2 x=1) ^(7 x≠1 ) ⇒ ∫_0 ^( 4) f(x)dx=?

$${f}\left({x}\right)=\left\{_{\mathrm{2}\:\:\:\:\:\:\:\:{x}=\mathrm{1}} ^{\mathrm{7}\:\:\:\:\:\:\:\:{x}\neq\mathrm{1}\:\:\:\:\:} \Rightarrow\:\int_{\mathrm{0}} ^{\:\mathrm{4}} {f}\left({x}\right){dx}=?\right. \\ $$

Question Number 203047    Answers: 0   Comments: 0

Question Number 202930    Answers: 1   Comments: 0

Question Number 202882    Answers: 3   Comments: 0

⇃

$$\:\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 202636    Answers: 0   Comments: 0

((^3 (√4^(5−x) ))/(∫_4 ^6 (x−1)dx)) = (1/2^(2x−1) ) , find the value of x. Solution (4^((5−x)/3) /(∫_4 ^6 ((x^2 /2)−x+k))) = (1/2^(2x−1) ) (2^(2•((5−x)/3)) /(((6^2 /2)−6+k)−((4^2 /2)−4+k))) = (1/2^(2x−1) ) (2^((10−2x)/3) /(((36)/2)−6+k−((16)/2)+4−k)) = (1/2^(2x−1) ) (2^((10−2x)/3) /(18−6−8+4)) = (1/2^(2x−1) ) (2^((10−2x)/3) /8) = (1/2^(2x−1) ) (Cross Multiply) 2^(2x−1) ×2^((10−2x)/3) = 8×1 2^(2x−1) ×2^((10−2x)/3) = 2^3 2^(2x−1+((10−2x)/3)) = 2^3 (Since, the bases are equal. Then, we can equate the exponents) 2x−1+((10−2x)/3) = 3 (Multiply each term by 3) 3(2x)−3(1)+3(((10−2x)/3)) = 3(3) 6x−3+10−2x = 9 (Collect Like Terms) 4x+7 = 9 4x = 9−7 4x = 2 (Divide Both Sides by 4) ((4x)/4) = (2/4) ∴ x = (1/2)

$$\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\:^{\mathrm{3}} \sqrt{\mathrm{4}^{\mathrm{5}−\mathrm{x}} }}{\int_{\mathrm{4}} ^{\mathrm{6}} \left(\mathrm{x}−\mathrm{1}\right){dx}}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} }\:,\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underline{\mathrm{Solution}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{4}^{\frac{\mathrm{5}−\mathrm{x}}{\mathrm{3}}} }{\int_{\mathrm{4}} ^{\mathrm{6}} \left(\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{2}}−\mathrm{x}+\mathrm{k}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\mathrm{2}\bullet\frac{\mathrm{5}−\mathrm{x}}{\mathrm{3}}} }{\left(\frac{\mathrm{6}^{\mathrm{2}} }{\mathrm{2}}−\mathrm{6}+\mathrm{k}\right)−\left(\frac{\mathrm{4}^{\mathrm{2}} }{\mathrm{2}}−\mathrm{4}+\mathrm{k}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} }{\frac{\mathrm{36}}{\mathrm{2}}−\mathrm{6}+\cancel{\mathrm{k}}−\frac{\mathrm{16}}{\mathrm{2}}+\mathrm{4}−\cancel{\mathrm{k}}}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} }{\mathrm{18}−\mathrm{6}−\mathrm{8}+\mathrm{4}}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}^{\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} }{\mathrm{8}}\:=\:\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2x}−\mathrm{1}} }\:\:\left(\mathrm{Cross}\:\mathrm{Multiply}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{2x}−\mathrm{1}} ×\mathrm{2}^{\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} \:=\:\mathrm{8}×\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{2x}−\mathrm{1}} ×\mathrm{2}^{\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} \:=\:\mathrm{2}^{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}^{\mathrm{2x}−\mathrm{1}+\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}} \:=\:\mathrm{2}^{\mathrm{3}} \:\left(\mathrm{Since},\:\mathrm{the}\:\mathrm{bases}\:\mathrm{are}\:\mathrm{equal}.\:\mathrm{Then},\:\mathrm{we}\:\mathrm{can}\:\mathrm{equate}\:\mathrm{the}\:\mathrm{exponents}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2x}−\mathrm{1}+\frac{\mathrm{10}−\mathrm{2x}}{\mathrm{3}}\:=\:\mathrm{3}\:\left(\mathrm{Multiply}\:\mathrm{each}\:\mathrm{term}\:\mathrm{by}\:\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\left(\mathrm{2x}\right)−\mathrm{3}\left(\mathrm{1}\right)+\cancel{\mathrm{3}}\left(\frac{\mathrm{10}−\mathrm{2x}}{\cancel{\mathrm{3}}}\right)\:=\:\mathrm{3}\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{6x}−\mathrm{3}+\mathrm{10}−\mathrm{2x}\:=\:\mathrm{9}\:\left(\mathrm{Collect}\:\mathrm{Like}\:\mathrm{Terms}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4x}+\mathrm{7}\:=\:\mathrm{9} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4x}\:=\:\mathrm{9}−\mathrm{7} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4x}\:=\:\mathrm{2}\:\left(\mathrm{Divide}\:\mathrm{Both}\:\mathrm{Sides}\:\mathrm{by}\:\mathrm{4}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\cancel{\mathrm{4}x}}{\cancel{\mathrm{4}}}\:=\:\frac{\mathrm{2}}{\mathrm{4}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\:\mathrm{x}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 202592    Answers: 0   Comments: 0

If I_n denotes ∫z^n e^(1/z) dz, then show that (n+1)!I_n =I_0 +e^(1/z) (1∙!z^2 +2∙!z^3 +∙∙∙+n!∙z^(n+1) )

$$\:\boldsymbol{{If}}\:\:\boldsymbol{{I}}_{\boldsymbol{{n}}} \:\boldsymbol{{denotes}}\:\int\boldsymbol{{z}}^{\boldsymbol{{n}}} \boldsymbol{{e}}^{\frac{\mathrm{1}}{\boldsymbol{{z}}}} \boldsymbol{{dz}},\:\boldsymbol{{then}}\:\boldsymbol{{show}}\:\boldsymbol{{that}} \\ $$$$\left(\boldsymbol{{n}}+\mathrm{1}\right)!\boldsymbol{{I}}_{\boldsymbol{{n}}} =\boldsymbol{{I}}_{\mathrm{0}} +\boldsymbol{{e}}^{\frac{\mathrm{1}}{\boldsymbol{{z}}}} \left(\mathrm{1}\centerdot!\boldsymbol{{z}}^{\mathrm{2}} +\mathrm{2}\centerdot!\boldsymbol{{z}}^{\mathrm{3}} +\centerdot\centerdot\centerdot+\boldsymbol{{n}}!\centerdot\boldsymbol{{z}}^{\boldsymbol{{n}}+\mathrm{1}} \right) \\ $$$$ \\ $$

Question Number 202591    Answers: 1   Comments: 0

Question Number 202543    Answers: 1   Comments: 1

Question Number 202522    Answers: 0   Comments: 0

Question Number 202490    Answers: 2   Comments: 2

Question Number 202485    Answers: 0   Comments: 3

Question Number 202448    Answers: 2   Comments: 0

Question Number 202418    Answers: 1   Comments: 0

Hard integral ∫∫∫∫∫∫∫∫∫ determinant ((a,b,c),(f,g,h),(j,k,l))dl dk dj dh dg df dc db da=

$$\mathrm{Hard}\:\mathrm{integral} \\ $$$$\int\int\int\int\int\int\int\int\int\begin{vmatrix}{{a}}&{{b}}&{{c}}\\{{f}}&{{g}}&{{h}}\\{{j}}&{{k}}&{{l}}\end{vmatrix}{dl}\:{dk}\:{dj}\:{dh}\:{dg}\:{df}\:{dc}\:{db}\:{da}= \\ $$

Question Number 202415    Answers: 2   Comments: 0

The value of ∫g′(x)f′(g(x))dx is...

$$\mathrm{The}\:\mathrm{value}\:\mathrm{of}\:\int{g}'\left({x}\right){f}'\left({g}\left({x}\right)\right){dx}\:\mathrm{is}... \\ $$

Question Number 202406    Answers: 2   Comments: 0

Question Number 202388    Answers: 1   Comments: 0

P rove that: ∫ (dx/(b^4 +2ax^2 +c))=((tan^(−1) ((((√2)(√a)x)/( (√(c+b^4 ))))))/( (√2)(√a)(√(c+b^4 ))))+C if a∙(c+b^4 )>0

$$\:\:\boldsymbol{{P}}\:\boldsymbol{{rove}}\:\boldsymbol{{that}}:\:\:\:\:\int\:\frac{\boldsymbol{{dx}}}{\boldsymbol{{b}}^{\mathrm{4}} +\mathrm{2}\boldsymbol{{ax}}^{\mathrm{2}} +\boldsymbol{{c}}}=\frac{\boldsymbol{{tan}}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{2}}\sqrt{\boldsymbol{{a}}}\boldsymbol{{x}}}{\:\sqrt{\boldsymbol{{c}}+\boldsymbol{{b}}^{\mathrm{4}} }}\right)}{\:\sqrt{\mathrm{2}}\sqrt{\boldsymbol{{a}}}\sqrt{\boldsymbol{{c}}+\boldsymbol{{b}}^{\mathrm{4}} }}+\boldsymbol{{C}} \\ $$$$\boldsymbol{{if}}\:\:\boldsymbol{{a}}\centerdot\left(\boldsymbol{{c}}+\boldsymbol{{b}}^{\mathrm{4}} \right)>\mathrm{0} \\ $$$$ \\ $$

Question Number 202212    Answers: 1   Comments: 1

Question Number 202167    Answers: 2   Comments: 0

∫^1 _0 ∫^1 _x sin(y^2 )dydx = ¿

$$\underset{\mathrm{0}} {\int}^{\mathrm{1}} \underset{{x}} {\int}^{\mathrm{1}} {sin}\left({y}^{\mathrm{2}} \right){dydx}\:=\:¿ \\ $$

Question Number 202127    Answers: 1   Comments: 0

Question Number 202125    Answers: 1   Comments: 0

∫ ((sin(3x))/(1+sin^3 x))dx

$$\int\:\frac{\boldsymbol{{sin}}\left(\mathrm{3}\boldsymbol{{x}}\right)}{\mathrm{1}+\boldsymbol{{sin}}^{\mathrm{3}} \boldsymbol{{x}}}\boldsymbol{{dx}} \\ $$

Question Number 201925    Answers: 0   Comments: 2

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