What is the integral of-5x(4-e^{x^2}) ?

The integral of-5x(4-e^{x^2}) is -5(2x^2-1/2 e^{x^2})+C

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integral of-5x(4-e^{x^2})

\int - 5 x (4 - e^{x^{2}}) d x

- 5 (2 x^{2} - \frac{1}{2} e^{x^{2}}) + C

Solution steps

\int - 5 x (4 - e^{x^{2}}) d x

Take the constant out:

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

= - 5 \cdot \int x (4 - e^{x^{2}}) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (4 - e^{x^{2}}) - \int - e^{x^{2}} x^{3} d x

= - 5 (\frac{1}{2} x^{2} (4 - e^{x^{2}}) - \int - e^{x^{2}} x^{3} d x)

\int - e^{x^{2}} x^{3} d x = - \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}})

= - 5 (\frac{1}{2} x^{2} (4 - e^{x^{2}}) - (- \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}})))

Simplify

- 5 (\frac{1}{2} x^{2} (4 - e^{x^{2}}) - (- \frac{1}{2} (e^{x^{2}} x^{2} - e^{x^{2}}))) : - 5 (2 x^{2} - \frac{1}{2} e^{x^{2}})

= - 5 (2 x^{2} - \frac{1}{2} e^{x^{2}})

Add a constant to the solution

= - 5 (2 x^{2} - \frac{1}{2} e^{x^{2}}) + C

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What is the integral of-5x(4-e^{x^2}) ?
The integral of-5x(4-e^{x^2}) is -5(2x^2-1/2 e^{x^2})+C