integral of (x-1)e^{x^2-2x} — step-by-step solution

⚠ 2x is read as the product 2*x: for a power, write 2^x‖(x-1)e^{x^2-2x} is read as (x-1)*e^{x^2-2x}: use * for multiplication, use ^ for a power‖(x-1)*e^{x^2-2x} is read as (x-1)*e^(x^2-2x)

Before integrating, let's calculate:

(x - 1) \cdot e^{(x^{2} - 2 x)}

x e^{(x^{2} - 2 x)} - e^{(x^{2} - 2 x)}

Let's integrate:

\int(x e^{(x^{2} - 2 x)} - e^{(x^{2} - 2 x)})dx

For the math expression:

x e^{(x^{2} - 2 x)}

, we know the derivative formula:

\frac{d}{d x} (\exp (a)) = \exp (a)

, so we try differentiating with exp

Let's differentiate:

\frac{d}{d x} (\exp (x^{2} - 2 x))

We differentiate using:

\frac{d}{d x} (\exp (f)) = \frac{d}{d x} (f) \cdot \exp (f)

(where f =

x^{2} - 2 x

)

\frac{d}{d x} (x^{2} - 2 x) \cdot \exp (x^{2} - 2 x)

(\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x)) \cdot \exp (x^{2} - 2 x)

ℹ

(\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 2 x)) \cdot \exp (x^{2} - 2 x)

\frac{d}{d x} (x^{2}) \cdot \exp (x^{2} - 2 x) + \frac{d}{d x} (- 2 x) \cdot \exp (x^{2} - 2 x)

‖

\frac{d}{d x} (x^{2})

2 x

‖

\frac{d}{d x} (- 2 x)

-2

2 x \cdot \exp (x^{2} - 2 x) - 2 \exp (x^{2} - 2 x)

You observe that the computed derivative and the original math expression are the same, except for a coefficient. So the integral is:

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

\frac{e^{(x^{2} - 2 x)}}{2}

▶Indefinite integrals are defined up to an additive constant C, so this yields:

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