What is the integral of e^{x+5}(x^2+3) ?

The integral of e^{x+5}(x^2+3) is e^5(e^xx^2-2e^xx+5e^x)+C

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integral of e^{x+5}(x^2+3)

\int e^{x + 5} (x^{2} + 3) d x

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

Solution steps

\int e^{x + 5} (x^{2} + 3) d x

e^{x + 5} (x^{2} + 3) = e^{5} e^{x} (x^{2} + 3)

= \int e^{5} e^{x} (x^{2} + 3) d x

Take the constant out:

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

= e^{5} \cdot \int e^{x} (x^{2} + 3) d x

Apply Integration By Parts

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

= e^{5} ((x^{2} + 3) e^{x} - \int 2 x e^{x} d x)

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

= e^{5} ((x^{2} + 3) e^{x} - 2 (e^{x} x - e^{x}))

Expand

(x^{2} + 3) e^{x} - 2 (e^{x} x - e^{x}) : e^{x} x^{2} - 2 e^{x} x + 5 e^{x}

= e^{5} (e^{x} x^{2} - 2 e^{x} x + 5 e^{x})

Add a constant to the solution

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

\frac{d}{d x} ((x^{2} + 5 x + 1) (x^{3} - 2 x))

y^{^{'}} + 4 y = - 1

\frac{d y}{d x} = \frac{y - y}{tan ( x )}

\int 8 sec^{3} (5 x) d x

x \to 7 + lim (\frac{5}{x - 7})

What is the integral of e^{x+5}(x^2+3) ?
The integral of e^{x+5}(x^2+3) is e^5(e^xx^2-2e^xx+5e^x)+C