What is the integral from 1 to 2 of (x^2+2x+4)/x ?

The integral from 1 to 2 of (x^2+2x+4)/x is 4ln(2)+7/2

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integral from 1 to 2 of (x^2+2x+4)/x

\int_{1}^{2} \frac{x ^{2} + 2 x + 4}{x} d x

4 ln (2) + \frac{7}{2}

Decimal

6.27258 \dots

Solution steps

\int_{1}^{2} \frac{x ^{2} + 2 x + 4}{x} d x

Expand

\frac{x ^{2} + 2 x + 4}{x} : x + 2 + \frac{4}{x}

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

\int_{1}^{2} x d x = \frac{3}{2}

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

\int_{1}^{2} \frac{4}{x} d x = 4 ln (2)

= \frac{3}{2} + 2 + 4 ln (2)

Combine the fractions

2 + \frac{3}{2} : \frac{7}{2}

= 4 ln (2) + \frac{7}{2}

What is the integral from 1 to 2 of (x^2+2x+4)/x ?
The integral from 1 to 2 of (x^2+2x+4)/x is 4ln(2)+7/2