What is the integral of (9ln^2(x)+8ln(x))/x ?

The integral of (9ln^2(x)+8ln(x))/x is 3ln^3(x)+4ln^2(x)+C

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integral of (9ln^2(x)+8ln(x))/x

\int \frac{9 ln ^{2} ( x ) + 8 ln ( x )}{x} d x

3 ln^{3} (x) + 4 ln^{2} (x) + C

Solution steps

\int \frac{9 ln ^{2} ( x ) + 8 ln ( x )}{x} d x

Expand

\frac{9 ln ^{2} ( x ) + 8 ln ( x )}{x} : \frac{9 ln ^{2} ( x )}{x} + \frac{8 ln ( x )}{x}

= \int \frac{9 ln ^{2} ( x )}{x} + \frac{8 ln ( x )}{x} 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 \frac{9 ln ^{2} ( x )}{x} d x + \int \frac{8 ln ( x )}{x} d x

\int \frac{9 ln ^{2} ( x )}{x} d x = 3 ln^{3} (x)

\int \frac{8 ln ( x )}{x} d x = 4 ln^{2} (x)

= 3 ln^{3} (x) + 4 ln^{2} (x)

Add a constant to the solution

= 3 ln^{3} (x) + 4 ln^{2} (x) + C

\frac{\partial}{\partial x} (4 x)

f (x) = 1 + x

\int 2 x + sin (x) d x

x^{- 2}, 5

\frac{d}{d x} (sin (x^{12}))

What is the integral of (9ln^2(x)+8ln(x))/x ?
The integral of (9ln^2(x)+8ln(x))/x is 3ln^3(x)+4ln^2(x)+C