What is the integral of x^2e^{x^{3/2}} ?

The integral of x^2e^{x^{3/2}} is 2/3 (e^{x^{3/2}}x^{3/2}-e^{x^{3/2}})+C

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integral of x^2e^{x^{3/2}}

\int x^{2} e^{x^{\frac{3}{2}}} d x

\frac{2}{3} (e^{x^{\frac{3}{2}}} x^{\frac{3}{2}} - e^{x^{\frac{3}{2}}}) + C

Solution steps

\int x^{2} e^{x^{\frac{3}{2}}} d x

Apply u-substitution

: \int \frac{2 e ^{u} u}{3} d u

= \int \frac{2 e ^{u} u}{3} d u

Take the constant out:

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

= \frac{2}{3} \cdot \int e^{u} u d u

Apply Integration By Parts

: e^{u} u - \int e^{u} d u

= \frac{2}{3} (e^{u} u - \int e^{u} d u)

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

= \frac{2}{3} (e^{u} u - e^{u})

Substitute back

u = x^{\frac{3}{2}}

= \frac{2}{3} (e^{x^{\frac{3}{2}}} x^{\frac{3}{2}} - e^{x^{\frac{3}{2}}})

Add a constant to the solution

= \frac{2}{3} (e^{x^{\frac{3}{2}}} x^{\frac{3}{2}} - e^{x^{\frac{3}{2}}}) + C

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What is the integral of x^2e^{x^{3/2}} ?
The integral of x^2e^{x^{3/2}} is 2/3 (e^{x^{3/2}}x^{3/2}-e^{x^{3/2}})+C