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 12047 by Joel576 last updated on 10/Apr/17

∫4x^3 (3x^2  + 2)^5  dx

$$\int\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{3}{x}^{\mathrm{2}} \:+\:\mathrm{2}\right)^{\mathrm{5}} \:{dx} \\ $$

Commented by Joel576 last updated on 10/Apr/17

Answered by sma3l2996 last updated on 10/Apr/17

t=3x^2 +2⇒dt=6xdx  x^3 dx=x^2 xdx=(((t−2)dt)/(3×6))  A=∫4x^3 (3x^2 +2)^5 dx=(2/9)∫t^5 (t−2)dt=(2/9)∫(t^6 −2t^5 )dt  =(2/9)((t^7 /7)−(t^6 /3))+C=(2/9)t^6 (((3t−7)/(21)))+C  A=(2/(189))(3x^2 +2)^6 (9x^2 −1)+C

$${t}=\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}\Rightarrow{dt}=\mathrm{6}{xdx} \\ $$$${x}^{\mathrm{3}} {dx}={x}^{\mathrm{2}} {xdx}=\frac{\left({t}−\mathrm{2}\right){dt}}{\mathrm{3}×\mathrm{6}} \\ $$$${A}=\int\mathrm{4}{x}^{\mathrm{3}} \left(\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{5}} {dx}=\frac{\mathrm{2}}{\mathrm{9}}\int{t}^{\mathrm{5}} \left({t}−\mathrm{2}\right){dt}=\frac{\mathrm{2}}{\mathrm{9}}\int\left({t}^{\mathrm{6}} −\mathrm{2}{t}^{\mathrm{5}} \right){dt} \\ $$$$=\frac{\mathrm{2}}{\mathrm{9}}\left(\frac{{t}^{\mathrm{7}} }{\mathrm{7}}−\frac{{t}^{\mathrm{6}} }{\mathrm{3}}\right)+{C}=\frac{\mathrm{2}}{\mathrm{9}}{t}^{\mathrm{6}} \left(\frac{\mathrm{3}{t}−\mathrm{7}}{\mathrm{21}}\right)+{C} \\ $$$${A}=\frac{\mathrm{2}}{\mathrm{189}}\left(\mathrm{3}{x}^{\mathrm{2}} +\mathrm{2}\right)^{\mathrm{6}} \left(\mathrm{9}{x}^{\mathrm{2}} −\mathrm{1}\right)+{C} \\ $$

Commented by Joel576 last updated on 11/Apr/17

thank you very much

$${thank}\:{you}\:{very}\:{much} \\ $$

Commented by sma3l2996 last updated on 11/Apr/17

you welcome

$${you}\:{welcome} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com