What is the integral of 3*e^x*(x^2+4) ?

The integral of 3*e^x*(x^2+4) is 3(e^xx^2-2e^xx+6e^x)+C

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integral of 3e^x(x^2+4)

\int 3 \cdot e^{x} \cdot (x^{2} + 4) d x

3 (e^{x} x^{2} - 2 e^{x} x + 6 e^{x}) + C

Solution steps

\int 3 e^{x} (x^{2} + 4) d x

Take the constant out:

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

= 3 \cdot \int e^{x} (x^{2} + 4) d x

Apply Integration By Parts

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

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

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

= 3 ((x^{2} + 4) e^{x} - 2 (e^{x} x - e^{x}))

Expand

(x^{2} + 4) e^{x} - 2 (e^{x} x - e^{x}) : e^{x} x^{2} - 2 e^{x} x + 6 e^{x}

= 3 (e^{x} x^{2} - 2 e^{x} x + 6 e^{x})

Add a constant to the solution

= 3 (e^{x} x^{2} - 2 e^{x} x + 6 e^{x}) + C

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What is the integral of 3*e^x*(x^2+4) ?
The integral of 3*e^x*(x^2+4) is 3(e^xx^2-2e^xx+6e^x)+C