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

The integral of e^{(x/3)^3}x^2 is 9e^{(x^3)/(27)}+C

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

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

9 e^{\frac{x ^{3}}{27}} + C

Solution steps

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

Simplify

e^{(\frac{x}{3})^{3}} x^{2} : e^{\frac{x ^{3}}{27}} x^{2}

= \int e^{\frac{x ^{3}}{27}} x^{2} d x

Apply u-substitution

: \int \frac{e ^{\frac{u}{27}}}{3} d u

= \int \frac{e ^{\frac{u}{27}}}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int e^{\frac{u}{27}} d u

Apply u-substitution

: \int e^{v} \cdot 27 d v

= \frac{1}{3} \cdot \int e^{v} \cdot 27 d v

Take the constant out:

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

= \frac{1}{3} \cdot 27 \cdot \int e^{v} d v

Use the common integral:

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

= \frac{1}{3} \cdot 27 e^{v}

Substitute back

= \frac{1}{3} \cdot 27 e^{\frac{x ^{3}}{27}}

Simplify

\frac{1}{3} \cdot 27 e^{\frac{x ^{3}}{27}} : 9 e^{\frac{x ^{3}}{27}}

= 9 e^{\frac{x ^{3}}{27}}

Add a constant to the solution

= 9 e^{\frac{x ^{3}}{27}} + C

\int \frac{- 2 x ^{2}}{9 - x ^{2}} d x

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\int u (u^{2} + 3 u + 7) d u

What is the integral of e^{(x/3)^3}x^2 ?
The integral of e^{(x/3)^3}x^2 is 9e^{(x^3)/(27)}+C