What is the integral of ((ln(x))^2+1)/(xln(x)) ?

The integral of ((ln(x))^2+1)/(xln(x)) is (ln^2(x))/2+ln|ln(x)|+C

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integral of ((ln(x))^2+1)/(xln(x))

\int \frac{( ln ( x ) ) ^{2} + 1}{x ln ( x )} d x

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

Solution steps

\int \frac{( ln ( x ) ) ^{2} + 1}{x ln ( x )} d x

Expand

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

= \int \frac{ln ( x )}{x} + \frac{1}{x ln ( 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{ln ( x )}{x} d x + \int \frac{1}{x ln ( x )} d x

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

\int \frac{1}{x ln ( x )} d x = ln ∣ ln (x) ∣

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

Add a constant to the solution

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

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\frac{d}{d x} (\frac{ln ( 3 x )}{x})

3 y^{^{''}} + 13 y^{^{'}} + 4 = 0, y (0) = 0, y^{^{'}} (0) = 0

\frac{d}{d x} (3 (7 - 3 x^{3})^{3})

- (1 + y^{2})^{\frac{1}{2}} d x + 4 x y d y = 0

What is the integral of ((ln(x))^2+1)/(xln(x)) ?
The integral of ((ln(x))^2+1)/(xln(x)) is (ln^2(x))/2+ln|ln(x)|+C