What is the integral of 2x(7-e^{(x^2)/4}) ?

The integral of 2x(7-e^{(x^2)/4}) is 7x^2-4e^{(x^2)/4}+C

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integral of 2x(7-e^{(x^2)/4})

\int 2 x (7 - e^{\frac{x ^{2}}{4}}) d x

7 x^{2} - 4 e^{\frac{x ^{2}}{4}} + C

Solution steps

\int 2 x (7 - e^{\frac{x ^{2}}{4}}) d x

Take the constant out:

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

= 2 \cdot \int x (7 - e^{\frac{x ^{2}}{4}}) d x

Apply Integration By Parts

: \frac{1}{2} x^{2} (7 - e^{\frac{x ^{2}}{4}}) - \int - \frac{1}{4} e^{\frac{x ^{2}}{4}} x^{3} d x

= 2 (\frac{1}{2} x^{2} (7 - e^{\frac{x ^{2}}{4}}) - \int - \frac{1}{4} e^{\frac{x ^{2}}{4}} x^{3} d x)

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

= 2 (\frac{1}{2} x^{2} (7 - e^{\frac{x ^{2}}{4}}) - (- \frac{1}{2} e^{\frac{x ^{2}}{4}} (x^{2} - 4)))

Simplify

2 (\frac{1}{2} x^{2} (7 - e^{\frac{x ^{2}}{4}}) - (- \frac{1}{2} e^{\frac{x ^{2}}{4}} (x^{2} - 4))) : 7 x^{2} - 4 e^{\frac{x ^{2}}{4}}

= 7 x^{2} - 4 e^{\frac{x ^{2}}{4}}

Add a constant to the solution

= 7 x^{2} - 4 e^{\frac{x ^{2}}{4}} + C

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What is the integral of 2x(7-e^{(x^2)/4}) ?
The integral of 2x(7-e^{(x^2)/4}) is 7x^2-4e^{(x^2)/4}+C