What is the integral of (1-x)(e-x+1) ?

The integral of (1-x)(e-x+1) is -(ex^2)/2-x^2+(x^3)/3+ex+x+C

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integral of (1-x)(e-x+1)

\int (1 - x) (e - x + 1) d x

- \frac{e x ^{2}}{2} - x^{2} + \frac{x ^{3}}{3} + e x + x + C

Solution steps

\int (1 - x) (e - x + 1) d x

Expand

(1 - x) (e - x + 1) : - e x - 2 x + x^{2} + e + 1

= \int - e x - 2 x + x^{2} + e + 1 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int e x d x - \int 2 x d x + \int x^{2} d x + \int e d x + \int 1 d x

\int e x d x = \frac{e x ^{2}}{2}

\int 2 x d x = x^{2}

\int x^{2} d x = \frac{x ^{3}}{3}

\int e d x = e x

\int 1 d x = x

= - \frac{e x ^{2}}{2} - x^{2} + \frac{x ^{3}}{3} + e x + x

Add a constant to the solution

= - \frac{e x ^{2}}{2} - x^{2} + \frac{x ^{3}}{3} + e x + x + C

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

x \to \infty lim (π 5^{x})

y = x^{2} - 2 x, y = - x^{4}

y = 7 x^{3} + 2

\int (t + 200)^{2} d t

What is the integral of (1-x)(e-x+1) ?
The integral of (1-x)(e-x+1) is -(ex^2)/2-x^2+(x^3)/3+ex+x+C