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

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

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integral from 1 to e of (3(ln(x))^2+x)/x

\int_{1}^{e} \frac{3 ( ln ( x ) ) ^{2} + x}{x} d x

e

Decimal

2.71828 \dots

Solution steps

\int_{1}^{e} \frac{3 ( ln ( x ) ) ^{2} + x}{x} d x

Simplify

= \int_{1}^{e} \frac{3 ln ^{2} ( x ) + x}{x} d x

Expand

\frac{3 ln ^{2} ( x ) + x}{x} : \frac{3 ln ^{2} ( x )}{x} + 1

= \int_{1}^{e} \frac{3 ln ^{2} ( x )}{x} + 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_{1}^{e} \frac{3 ln ^{2} ( x )}{x} d x + \int_{1}^{e} 1 d x

\int_{1}^{e} \frac{3 ln ^{2} ( x )}{x} d x = 1

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

= 1 + e - 1

Simplify

= e

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