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

The integral of ((ln(x))^n)/x is (ln^{n+1}(x))/(n+1)+C

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

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

\frac{ln ^{n + 1} ( x )}{n + 1} + C

Solution steps

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

Simplify

= \int \frac{ln ^{n} ( x )}{x} d x

Apply u-substitution

: \int u^{n} d u

= \int u^{n} d u

Apply the Power Rule

: \frac{u ^{n + 1}}{n + 1}

= \frac{u ^{n + 1}}{n + 1}

Substitute back

u = ln (x)

= \frac{ln ^{n + 1} ( x )}{n + 1}

Add a constant to the solution

= \frac{ln ^{n + 1} ( x )}{n + 1} + C

y = x^{2} - x + 7, at x = 2

\int_{- 2}^{2} \frac{1}{36 + x ^{2}} d x

2 y^{^{''}} + 5 y^{^{'}} - 3 y = 0, y (0) = 1, y^{^{'}} (0) = 25

\frac{\partial}{\partial x} (2 x y)

\int 4 x^{3} + 3 x^{2} d x

What is the integral of ((ln(x))^n)/x ?
The integral of ((ln(x))^n)/x is (ln^{n+1}(x))/(n+1)+C