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Question Number 75089 by MJS last updated on 07/Dec/19

$$\int \sin (x^3+c)dx=?$$$$\int \sinh (x^3+c)dx=?$$

Commented by mathmax by abdo last updated on 07/Dec/19

$$\int sin(x^3+c)dx=Im(\int e^{i(x^3+c)}dx=)and$$$$\int e^{i(x^3+c)}dx=e^{ic}\int e^{i{x}^3}dx=e^{ic}\int e^{-(-i{x}^3)}dx$$$$changement-i{x}^3=tgive{x}^3=it\Rightarrow x={\left(it\right)}^{\frac{1}{3}}\Rightarrow$$$$\int e^{i(x^3+c)}dx=\frac{1}{3}{e}^{ic}\int e^{-t}{\left(it\right)}^{\frac{1}{3}-1}dt$$$$=\frac{1}{3}{e}^{ic}{i}^{\frac{1}{3}-1}\int t^{\frac{1}{3}-1}{e}^{-t}dt$$$$=\frac{1}{3}{e}^{ic}{\left(e^{\frac{i\pi }{2}}\right)}^{-\frac{2}{3}}\int t^{\frac{1}{3}-1}{e}^{-t}dt$$$$=\frac{1}{3}{e}^{ic-3i\pi }{\mathrm{\Gamma }}_{in}\left(\frac{1}{3}\right)=\frac{1}{3}{e}^{i(c-3\pi )}{\mathrm{\Gamma }}_{in}\left(\frac{1}{3}\right)$$$${\mathrm{\Gamma }}_{in}istheincomplet\mathrm{\Gamma }....$$

Commented by mathmax by abdo last updated on 07/Dec/19

$$forgive\int e^{i(x^3+c)}dx=\frac{1}{3}{e}^{ic-i\frac{\pi }{3}}{\mathrm{\Gamma }}_{in}\left(\frac{1}{3}\right)\Rightarrow$$$$I=\frac{1}{3}{\mathrm{\Gamma }}_{in}\left(\frac{1}{3}\right)sin(c-\frac{\pi }{3})$$

Commented by MJS last updated on 07/Dec/19

$$\mathrm{thank}\mathrm{you}$$

Commented by mathmax by abdo last updated on 14/Dec/19

$$youarewelcomesir.$$

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