What is the integral from x^3 to x of e^{(2y)/x} ?

The integral from x^3 to x of e^{(2y)/x} is (x(e^2-e^{2x^2}))/2

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integral from x^3 to x of e^{(2y)/x}

\int_{x^{3}}^{x} e^{\frac{2 y}{x}} d y

\frac{x ( e ^{2} - e ^{2 x^{2}} )}{2}

Solution steps

\int_{x^{3}}^{x} e^{\frac{2 y}{x}} d y

Apply u-substitution

: \int_{2 x^{2}}^{2} \frac{x e ^{u}}{2} d u

= \int_{2 x^{2}}^{2} \frac{x e ^{u}}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{x}{2} \cdot \int_{2 x^{2}}^{2} e^{u} d u

Use the common integral:

\int e^{u} d u = e^{u}

= \frac{x}{2} [e^{u}]_{2 x^{2}}^{2}

Compute the boundaries:

e^{2} - e^{2 x^{2}}

= \frac{x}{2} (e^{2} - e^{2 x^{2}})

Simplify

= \frac{x ( e ^{2} - e ^{2 x^{2}} )}{2}

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What is the integral from x^3 to x of e^{(2y)/x} ?
The integral from x^3 to x of e^{(2y)/x} is (x(e^2-e^{2x^2}))/2