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

The integral of (x^2+3)e^{2x-3} is 1/4 e^{2x-3}(2x^2-2x+7)+C

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

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

\frac{1}{4} e^{2 x - 3} (2 x^{2} - 2 x + 7) + C

Solution steps

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

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

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

Take the constant out:

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

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

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

Apply Integration By Parts

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

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

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

= \frac{1}{e ^{3}} (\frac{1}{2} e^{2 x} (x^{2} + 3) - \frac{1}{4} (e^{2 x} \cdot 2 x - e^{2 x}))

Simplify

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

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

Add a constant to the solution

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

\int \frac{5 1 - 3 x}{4} d x

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\int 2 sin (a - \frac{x}{2}) d x

\int - 3.4 (4.5 - t) sin (0.4 (3 - t)) + 80 d t

\int 2 + 6 x d x

What is the integral of (x^2+3)e^{2x-3} ?
The integral of (x^2+3)e^{2x-3} is 1/4 e^{2x-3}(2x^2-2x+7)+C