What is the integral from 1 to e of (x^5-2x+1)/x ?

The integral from 1 to e of (x^5-2x+1)/x is 3+(e^5-1)/5-2e

Popular >

integral from 1 to e of (x^5-2x+1)/x

\int_{1}^{e} \frac{x ^{5} - 2 x + 1}{x} d x

3 + \frac{e ^{5} - 1}{5} - 2 e

Decimal Notation

27.04606 \dots

Solution steps

\int_{1}^{e} \frac{x ^{5} - 2 x + 1}{x} d x

Expand

\frac{x ^{5} - 2 x + 1}{x} : x^{4} - 2 + \frac{1}{x}

= \int_{1}^{e} x^{4} - 2 + \frac{1}{x} 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_{1}^{e} x^{4} d x - \int_{1}^{e} 2 d x + \int_{1}^{e} \frac{1}{x} d x

\int_{1}^{e} x^{4} d x = \frac{e ^{5} - 1}{5}

\int_{1}^{e} 2 d x = 2 e - 2

\int_{1}^{e} \frac{1}{x} d x = 1

= \frac{e ^{5} - 1}{5} - (2 e - 2) + 1

Simplify

= 3 + \frac{e ^{5} - 1}{5} - 2 e

\int_{0}^{2} x^{2} 4 - x d x

\int_{2}^{10} \frac{1}{1 - x ^{2}} d x

\int_{2}^{x^{3}} ln (t^{2}) d t

\int_{- π}^{π} sin (2 x) cos (3 x) d x

\int_{0}^{π} cos (6 x) sin (5 x) d x

What is the integral from 1 to e of (x^5-2x+1)/x ?
The integral from 1 to e of (x^5-2x+1)/x is 3+(e^5-1)/5-2e