What is the integral of 3x^2e^{-0.01x^3} ?

The integral of 3x^2e^{-0.01x^3} is -100e^{-0.01x^3}+C

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integral of 3x^2e^{-0.01x^3}

\int 3 x^{2} e^{- 0.01 x^{3}} d x

- 100 e^{- 0.01 x^{3}} + C

Solution steps

\int 3 x^{2} e^{- 0.01 x^{3}} d x

Take the constant out:

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

= 3 \cdot \int x^{2} e^{- 0.01 x^{3}} d x

Apply u-substitution

= 3 \cdot \int \frac{e ^{- 0.01 u}}{3} d u

Take the constant out:

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

= 3 \cdot \frac{1}{3} \cdot \int e^{- 0.01 u} d u

Apply u-substitution

= 3 \cdot \frac{1}{3} \cdot \int - \frac{1}{0.01} e^{v} d v

Take the constant out:

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

= 3 \cdot \frac{1}{3} (- \frac{1}{0.01} \cdot \int e^{v} d v)

Simplify

= 3 \cdot \frac{1}{3} (- 100 \cdot \int e^{v} d v)

Use the common integral:

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

= 3 \cdot \frac{1}{3} (- 100 e^{v})

Substitute back

= 3 \cdot \frac{1}{3} (- 100 e^{- 0.01 x^{3}})

Simplify

3 \cdot \frac{1}{3} (- 100 e^{- 0.01 x^{3}}) : - 100 e^{- 0.01 x^{3}}

= - 100 e^{- 0.01 x^{3}}

Add a constant to the solution

= - 100 e^{- 0.01 x^{3}} + C

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What is the integral of 3x^2e^{-0.01x^3} ?
The integral of 3x^2e^{-0.01x^3} is -100e^{-0.01x^3}+C