Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 10876 by Saham last updated on 28/Feb/17

∫_( 0) ^( 1) ∫_( x) ^( (√x))  (x + y^5 ) dy dx

$$\int_{\:\mathrm{0}} ^{\:\mathrm{1}} \int_{\:\mathrm{x}} ^{\:\sqrt{\mathrm{x}}} \:\left(\mathrm{x}\:+\:\mathrm{y}^{\mathrm{5}} \right)\:\mathrm{dy}\:\mathrm{dx} \\ $$

Answered by fariraihmudzengerere75@gmail.c last updated on 28/Feb/17

Answer . ∫_0 ^1  ∫_x ^(√x)   (x+y^5 )dydx  =∫_0 ^1  [  xy +(y^6 /6)+c ]_x ^(√x) dx   =∫_0 ^1  [x×(√(x ))  +((((√(x)^6     )))/6)−x^(2  ) −(x^6 /6)   +c−c]dx  =∫_0 ^1 [x^(3/2) + (x^3 /6) −x^2 −(x^6 /6)]dx  =[(x^(5/2) /(5/2))+(x^4 /(24)) −(x^3 /3) −(x^7 /(42))  +c−c]_0 ^1   =(2/5)+(1/(24))−(1/3)−(1/(42))  =0 .084  523  809

$${Answer}\:.\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\int_{{x}} ^{\sqrt{{x}}} \:\:\left({x}+{y}^{\mathrm{5}} \right){dydx} \\ $$ $$=\int_{\mathrm{0}} ^{\mathrm{1}} \:\left[\:\:{xy}\:+\frac{{y}^{\mathrm{6}} }{\mathrm{6}}+{c}\:\right]_{{x}} ^{\sqrt{{x}}} {dx}\: \\ $$ $$=\int_{\mathrm{0}} ^{\mathrm{1}} \:\left[{x}×\sqrt{{x}\:}\:\:+\frac{\left(\sqrt{\left.{x}\right)^{\mathrm{6}} \:\:\:\:}\right.}{\mathrm{6}}−{x}^{\mathrm{2}\:\:} −\frac{{x}^{\mathrm{6}} }{\mathrm{6}}\:\:\:+{c}−{c}\right]{dx} \\ $$ $$=\int_{\mathrm{0}} ^{\mathrm{1}} \left[{x}^{\mathrm{3}/\mathrm{2}} +\:\frac{{x}^{\mathrm{3}} }{\mathrm{6}}\:−{x}^{\mathrm{2}} −\frac{{x}^{\mathrm{6}} }{\mathrm{6}}\right]{dx} \\ $$ $$=\left[\frac{{x}^{\mathrm{5}/\mathrm{2}} }{\mathrm{5}/\mathrm{2}}+\frac{{x}^{\mathrm{4}} }{\mathrm{24}}\:−\frac{{x}^{\mathrm{3}} }{\mathrm{3}}\:−\frac{{x}^{\mathrm{7}} }{\mathrm{42}}\:\:+{c}−{c}\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$ $$=\frac{\mathrm{2}}{\mathrm{5}}+\frac{\mathrm{1}}{\mathrm{24}}−\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{42}} \\ $$ $$=\mathrm{0}\:.\mathrm{084}\:\:\mathrm{523}\:\:\mathrm{809} \\ $$

Commented bySaham last updated on 28/Feb/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com