Question

Integrate (cbrte^sin(1/x))/(x^2 sec(1/x))

Answer 1

To solve:

$\int\dfrac{{\sqrt[3]{\mathrm{e}^{\sin{(\frac{1}{x})}}}}}{x^2\sec(\frac{1}{x})}dx$

We can re-write the above as:

$\int \dfrac{\mathrm{e}^\frac{\sin\left(\frac{1}{x}\right)}{3}}{\sec\left(\frac{1}{x}\right)x^2} dx \quad = \int \dfrac{\cos(\frac{1}{x})\mathrm{e}^\frac{\sin\left(\frac{1}{x}\right)}{3}}{x^2} dx$

Substitute $u = \dfrac{sin(\frac{1}{x})}{3}$

$\therefore \quad \dfrac{du}{dx}= \dfrac{cos(\frac{1}{x})}{3x^2} \implies dx = -\dfrac{3x^2}{cos(\frac{1}{x})}du$

Since $u = \dfrac{sin(\frac{1}{x})}{3}$ , then we have: $-3\int \mathrm{e}^u du = -3\mathrm{e}^u \implies -3\mathrm{e}^\frac{sin(\frac{1}{x})}{3} + C$