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 91615 by  M±th+et+s last updated on 01/May/20

$$hieveryoneisitrightifweusetylor$$$$inthisintegrationandiftherewere$$$$anotherwaythatwillbeverycool$$$$\int sin\left(x^4\right)dx$$$$$$$$$$

Commented by MJS last updated on 02/May/20

$$\mathrm{you}\mathrm{can}\mathrm{use}\sin \theta =\frac{{\mathrm{e}}^{\mathrm{i}\theta }-{\mathrm{e}}^{-\mathrm{i}\theta }}{2\mathrm{i}}\mathrm{and}\mathrm{then}\mathrm{the}$$$$\mathrm{incomplete}\mathrm{\Gamma }-\mathrm{function}$$

Commented by  M±th+et+s last updated on 02/May/20

$$thankyousirmjs$$$$letmetry$$$$$$$$I=\int sin\left(x^4\right)dx=\int \frac{e^{{ix}^4}-e^{-{ix}^4}}{2i}dx$$$$=\frac{1}{2i}\int e^{{ix}^4}dx-\frac{1}{2i}\int e^{-{ix}^4}dx$$$${ix}^4=-k{ix}^4=p$$$$x=i^{\frac{1}{4}}{k}^{\frac{1}{4}}x=(-i^{\frac{1}{4}})p^{\frac{1}{4}}$$$$dx=\frac{i^{\frac{1}{4}}}{4}{k}^{\frac{-3}{4}}dkdx=\frac{(-i^{\frac{1}{4}})}{4}{p}^{-\frac{3}{4}}dp$$$$$$$$I=\frac{1}{2i}\int e^{-k}\frac{i^{\frac{1}{4}}}{4}{k}^{-\frac{3}{4}}dk-\frac{1}{2i}\int e^{-p}\frac{(-i^{\frac{1}{4}})}{4}{p}^{\frac{-3}{4}}dp$$$$I=\frac{i^{-\frac{3}{4}}}{8}\int e^{-k}{k}^{(-\frac{3}{4}+1)-1}dk-\frac{{(-1)}^{\frac{1}{4}}{\left(i\right)}^{\frac{-3}{4}}}{8}\int e^{-p}{p}^{\frac{1}{4}-1}dp$$$$I=\frac{{\left(i\right)}^{\frac{-3}{4}}}{8}[-\mathrm{\Gamma }(\frac{1}{4},k)]-\frac{{(-1)}^{\frac{1}{4}}{\left(i\right)}^{-\frac{3}{4}}}{8}[-\mathrm{\Gamma }(\frac{1}{4},p)]+c$$$$I=\frac{-{\left(i\right)}^{\frac{-3}{4}}}{8}\mathrm{\Gamma }(\frac{1}{4},-{ix}^4)+\frac{{(-1)}^{\frac{1}{4}}{\left(i\right)}^{\frac{-3}{4}}}{8}\mathrm{\Gamma }(\frac{1}{4},{ix}^4)+c$$$$$$$$\mathrm{\Gamma }(a,x)isincompletegammafunction$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com