What is the integral from 2 to 5 of (ln(x^5))/x ?

The integral from 2 to 5 of (ln(x^5))/x is (25ln^2(5)-25ln^2(2))/(10)

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integral from 2 to 5 of (ln(x^5))/x

\int_{2}^{5} \frac{ln ( x ^{5} )}{x} d x

\frac{25 ln ^{2} ( 5 ) - 25 ln ^{2} ( 2 )}{10}

Decimal Notation

5.27459 \dots

Solution steps

\int_{2}^{5} \frac{ln ( x ^{5} )}{x} d x

Apply u-substitution

: \int_{32}^{3125} \frac{ln ( u )}{5 u} d u

= \int_{32}^{3125} \frac{ln ( u )}{5 u} d u

Take the constant out:

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

= \frac{1}{5} \cdot \int_{32}^{3125} \frac{ln ( u )}{u} d u

Apply u-substitution

: \int_{5 l n (2)}^{5 l n (5)} v d v

= \frac{1}{5} \cdot \int_{5 l n (2)}^{5 l n (5)} v d v

Apply the Power Rule

: [\frac{v ^{2}}{2}]_{5 l n (2)}^{5 l n (5)}

= \frac{1}{5} [\frac{v ^{2}}{2}]_{5 l n (2)}^{5 l n (5)}

Compute the boundaries:

\frac{25 ln ^{2} ( 5 ) - 25 ln ^{2} ( 2 )}{2}

= \frac{1}{5} \cdot \frac{25 ln ^{2} ( 5 ) - 25 ln ^{2} ( 2 )}{2}

Simplify

= \frac{25 ln ^{2} ( 5 ) - 25 ln ^{2} ( 2 )}{10}

(1 + x)^{\frac{1}{3}}

\int 5 e^{2 x} d x

y^{^{''}} + 9 y = t e^{3 t}

y^{^{'}} - y = 8 e^{x} + 18 e^{4 x}

\int 4 sec^{2} (x) tan (x) d x

What is the integral from 2 to 5 of (ln(x^5))/x ?
The integral from 2 to 5 of (ln(x^5))/x is (25ln^2(5)-25ln^2(2))/(10)