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 148408 by cesarL last updated on 27/Jul/21

$$\int x^5{e}^{x^2}dx$$$$Helpplease!$$

Answered by Olaf_Thorendsen last updated on 27/Jul/21

$$\mathrm{F}\left(x\right)=\int x^5{e}^{x^2}dx$$$$\mathrm{F}\left(x\right)=\int \frac{1}{2}{x}^4\left(2{xe}^{x^2}\right)dx$$$$\mathrm{F}\left(x\right)=\frac{1}{2}{x}^4{e}^{x^2}-\int 2x^3{e}^{x^2}dx$$$$\mathrm{F}\left(x\right)=\frac{1}{2}{x}^4{e}^{x^2}-\int x^2\left(2{xe}^{x^2}\right)dx$$$$\mathrm{F}\left(x\right)=\frac{1}{2}{x}^4{e}^{x^2}-x^2{e}^{x^2}+\int 2{xe}^{x^2}dx$$$$\mathrm{F}\left(x\right)=\frac{1}{2}{x}^4{e}^{x^2}-x^2{e}^{x^2}+e^{x^2}+\mathrm{C}$$$$\mathrm{F}\left(x\right)=(\frac{1}{2}{x}^4-x^2+1)e^{x^2}+\mathrm{C}$$

Commented by cesarL last updated on 27/Jul/21

$$thankusir$$

Answered by Math_Freak last updated on 27/Jul/21

$$\int {\left(x^2\right)}^2{xe}^{x^2}dx$$$$letu=x^2$$$$dx=\frac{du}{2x}$$$$\int u^2{xe}^u\bullet \frac{du}{2x}$$$$\frac{1}{2}\int u^2{e}^{u}du$$$$integratingbyparts$$$$\frac{1}{2}(u^2{e}^u-2\int {ue}^{u}du)$$$$\frac{1}{2}(u^2{e}^u-2({ue}^u-\int e^{u}du))$$$$\frac{1}{2}(u^2{e}^u-2({ue}^u-e^u))+C$$$$\frac{u^2{e}^u}{2}-{ue}^u+e^u+C$$$$u=x^2$$$$\frac{x^4{e}^{x^2}}{2}-x^2{e}^{x^2}+e^{x^2}+C$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com