What is the integral of ((ln(x^7)))/x ?

The integral of ((ln(x^7)))/x is 7/2 ln^2(x)+C

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integral of ((ln(x^7)))/x

\int \frac{( ln ( x ^{7} ) )}{x} d x

\frac{7}{2} ln^{2} (x) + C

Solution steps

\int \frac{ln ( x ^{7} )}{x} d x

Apply u-substitution

: \int \frac{ln ( u )}{7 u} d u

= \int \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 \frac{ln ( u )}{u} d u

Apply u-substitution

: \int v d v

= \frac{1}{7} \cdot \int v d v

Apply the Power Rule

: \frac{v ^{2}}{2}

= \frac{1}{7} \cdot \frac{v ^{2}}{2}

Substitute back

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

Simplify

\frac{1}{7} \cdot \frac{ln ^{2} ( x ^{7} )}{2} : \frac{7}{2} ln^{2} (x)

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

Add a constant to the solution

= \frac{7}{2} ln^{2} (x) + C

y^{^{'}} = \frac{3 x ^{2} - e ^{x}}{2 y - 5}

\frac{\partial}{\partial x} (x^{2} + y^{2} + 1)

y = 4 ln (cos (x))

\int \frac{x ^{2} - 4}{x ^{4}} d x

\frac{x}{x + 6}

What is the integral of ((ln(x^7)))/x ?
The integral of ((ln(x^7)))/x is 7/2 ln^2(x)+C