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

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

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integral from 1 to 2 of (ln(x^7))/x

\int_{1}^{2} \frac{ln ( x ^{7} )}{x} d x

\frac{7 ln ^{2} ( 2 )}{2}

Decimal

1.68158 \dots

Solution steps

\int_{1}^{2} \frac{ln ( x ^{7} )}{x} d x

Apply u-substitution

= \int_{1}^{128} \frac{ln ( u )}{7 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{7} \cdot \int_{1}^{128} \frac{ln ( u )}{u} d u

Apply u-substitution

= \frac{1}{7} \cdot \int_{0}^{7 l n (2)} v d v

Apply the Power Rule

= \frac{1}{7} [\frac{v ^{2}}{2}]_{0}^{7 l n (2)}

Compute the boundaries:

\frac{49 ln ^{2} ( 2 )}{2}

= \frac{1}{7} \cdot \frac{49 ln ^{2} ( 2 )}{2}

Simplify

= \frac{7 ln ^{2} ( 2 )}{2}

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