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 212626 by Ghisom last updated on 19/Oct/24

$$\mathrm{let}f\left(x\right)=\frac{1}{\sqrt{(x-a)(x-b)(x-c)}}$$$$\mathrm{let}a,b,c\in \mathbb{R}\wedge a<b<c$$$$\Rightarrow D\left(f\left(x\right)\right)=(a,b)\cup (c,\mathrm{\infty })$$$$\mathrm{prove}\underset{a}{\stackrel{b}{\int }}f\left(x\right)dx=\underset{c}{\stackrel{\mathrm{\infty }}{\int }}f\left(x\right)dx$$

Answered by MrGaster last updated on 02/Nov/24

$${\int }_a^{b}f\left(x\right)dx={\int }_a^b\frac{1}{\sqrt{(x-a)(x-b)(x-c)}}dx$$$${\int }_c^{\mathrm{\infty }}f\left(x\right)dx={\int }_c^{\mathrm{\infty }}\frac{1}{\sqrt{(x-a)(x-b)(x-c)}}dxx=$$$$a+(b-a)tdx=(b-a)dt$$$${\int }_a^b\frac{1}{\sqrt{(x-a(x-b(x-c)}}dx={\int }_0^1\frac{(b-a)}{\sqrt{(b-a)(b-a)t(b-a)(1-t)(b-a)(1-\frac{a+(b-a)t-c}{b-a})}}dt$$$$={\int }_0^1\frac{1}{\sqrt{t(1-t)(1-\frac{a-c}{b-a}-t)}}dtx=c+(x-c)t^{\prime }dx=(x-c){dt}^{\prime }$$$${\int }_c^{\mathrm{\infty }}\frac{1}{\sqrt{(x-a)(x-b)(x-c)}}dx={\int }_0^{\mathrm{\infty }}\frac{(x-c)}{\sqrt{(x-c)(x-c)t^{\prime }(x-c)(1-t^{\prime })(x-c)}}{dt}^{\prime }$$$$$$$$={\int }_0^{\mathrm{\infty }}\frac{1}{\sqrt{t^{\prime }(1-t^{\prime })(1+\frac{x-c}{c-a}-t^{\prime })}}{dt}^{\prime }$$$$t^{\prime }=1-{tdt}^{\prime }=-dt$$$${\int }_0^{\mathrm{\infty }}\frac{1}{\sqrt{t^{\prime }(1-t^{\prime })(1+\frac{x-c}{c-a}-t^{\prime })}}{dt}^{\prime }={\int }_0^{-\mathrm{\infty }}\frac{-1}{\sqrt{(1-t)\left(t\right)(1+\frac{x-c}{c-a}-1+t)}}dt$$$$={\int }_{-\mathrm{\infty }}^1\frac{1}{\sqrt{(1-t)\left(t\right)(1+\frac{x-c}{c-a}-1+t)}}dt={\int }_{-\mathrm{\infty }}^1\frac{1}{\sqrt{(1-t)\left(t\right)(t+\frac{x-c}{c-a})}}dt={\int }_{-\mathrm{\infty }}^1\frac{1}{\sqrt{(1-t)\left(t\right)(t+\frac{b-c}{b-a})}}dt={\int }_0^1\frac{1}{\sqrt{t(1-t)(1-\frac{a-c}{b-a})}}dt$$$${\int }_a^{b}f\left(x\right)dx={\int }_c^{\mathrm{\infty }}f\left(x\right)dx$$$$$$

Commented by Ghisom last updated on 02/Nov/24

$$\mathrm{you}\mathrm{write}$$$$a+(b-a)tdx=(b-a)dt$$$$\mathrm{but}\mathrm{this}\mathrm{makes}\mathrm{no}\mathrm{sense}.\mathrm{please}\mathrm{correct}$$$$\mathrm{and}\mathrm{clarify}\mathrm{your}\mathrm{answer}.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com