Use \(u=x^{1/a}\) to reduce \[ \int e^{x^{1/a}}\,dx, \] and evaluate the special cases \[ \int e^{\sqrt x}\,dx, \qquad \int e^{\sqrt[3]x}\,dx. \] Also evaluate \[ \int e^x\sin x\,dx, \qquad \int \frac{\sin(\sqrt x)e^{\sqrt x}}{\sqrt x}\,dx, \qquad \int e^x\cos x\,dx. \]
Solution
With \(u=x^{1/a}\), so \(x=u^a\) and \(dx=a u^{a-1}du\), \[ \int e^{x^{1/a}}\,dx=a\int u^{a-1}e^u\,du. \] For positive integer \(a\) this is elementary by repeated integration by parts.
For \(a=2\), \[ \boxed{\int e^{\sqrt x}\,dx=2(\sqrt x-1)e^{\sqrt x}+C.} \] For \(a=3\), \[ \boxed{\int e^{\sqrt[3]x}\,dx=3\left(x^{2/3}-2x^{1/3}+2\right)e^{x^{1/3}}+C.} \]
Integration by parts twice gives \[ \boxed{\int e^x\sin x\,dx=\frac12e^x(\sin x-\cos x)+C,} \] \[ \boxed{\int e^x\cos x\,dx=\frac12e^x(\cos x+\sin x)+C.} \] Finally, putting \(u=\sqrt x\) gives \[ \int \frac{\sin(\sqrt x)e^{\sqrt x}}{\sqrt x}\,dx =2\int e^u\sin u\,du =\boxed{e^{\sqrt x}(\sin\sqrt x-\cos\sqrt x)+C.} \]