What is the integral of (2x^2-csc^2(ln(x)))/x ?

The integral of (2x^2-csc^2(ln(x)))/x is x^2-ln|x|+ln(x)+cot(ln(x))+C

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integral of (2x^2-csc^2(ln(x)))/x

\int \frac{2 x ^{2} - csc ^{2} ( ln ( x ) )}{x} d x

x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x)) + C

Solution steps

\int \frac{2 x ^{2} - csc ^{2} ( ln ( x ) )}{x} d x

Expand

\frac{2 x ^{2} - csc ^{2} ( ln ( x ) )}{x} : 2 x - \frac{csc ^{2} ( ln ( x ) )}{x}

= \int 2 x - \frac{csc ^{2} ( 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 2 x d x - \int \frac{csc ^{2} ( ln ( x ) )}{x} d x

\int 2 x d x = x^{2}

\int \frac{csc ^{2} ( ln ( x ) )}{x} d x = ln ∣ x ∣ - ln (x) - cot (ln (x))

= x^{2} - (ln ∣ x ∣ - ln (x) - cot (ln (x)))

Simplify

= x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x))

Add a constant to the solution

= x^{2} - ln ∣ x ∣ + ln (x) + cot (ln (x)) + C

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What is the integral of (2x^2-csc^2(ln(x)))/x ?
The integral of (2x^2-csc^2(ln(x)))/x is x^2-ln|x|+ln(x)+cot(ln(x))+C