What is ln|x+1|-1/2 ln|x^2+1|+arctan(x)+C ?

The solution to ln|x+1|-1/2 ln|x^2+1|+arctan(x)+C is ln((|x+1|sqrt(|x^2+1|))/(|x^2+1|))+arctan(x)+C

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simplify ln|x+1|-1/2 ln|x^2+1|+arctan(x)+C

simplify

ln ∣ x + 1 ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

ln (\frac{∣ x + 1 ∣ ∣ x ^{2} + 1 ∣}{∣ x ^{2} + 1 ∣}) + arctan (x) + C

Solution steps

ln ∣ x + 1 ∣ - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Apply log rule:

a lo g_{c} (b) = lo g_{c} (b^{a})

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

= ln ∣ x + 1 ∣ - ln (x^{2} + 1^{\frac{1}{2}}) + arctan (x) + C

Apply log rule:

lo g_{c} (a) - lo g_{c} (b) = lo g_{c} (\frac{a}{b})

ln ∣ x + 1 ∣ - ln (x^{2} + 1^{\frac{1}{2}}) = ln (\frac{∣ x + 1 ∣}{∣ x ^{2} + 1 ∣ ^{\frac{1}{2}}})

= ln (\frac{∣ x + 1 ∣}{∣ x ^{2} + 1 ∣ ^{\frac{1}{2}}}) + arctan (x) + C

Simplify

\frac{∣ x + 1 ∣}{∣ x ^{2} + 1 ∣ ^{\frac{1}{2}}} : \frac{∣ x + 1 ∣ ∣ x ^{2} + 1 ∣}{∣ x ^{2} + 1 ∣}

= ln (\frac{∣ x + 1 ∣ ∣ x ^{2} + 1 ∣}{∣ x ^{2} + 1 ∣}) + arctan (x) + C

ln (\frac{4 x}{x + 1})

4.7 + lo g_{10} (\frac{0.2}{0.1})

3 ln ∣ x - 3 ∣ + 5 ln ∣ x + 2 ∣ + \frac{5}{9} ln ∣ 2 x + 3 ∣

lo g_{10} (a) + 4 lo g_{10} (b) - 2 lo g_{10} (c)

lo g_{10} (\frac{a}{b ^{2}})

What is ln|x+1|-1/2 ln|x^2+1|+arctan(x)+C ?
The solution to ln|x+1|-1/2 ln|x^2+1|+arctan(x)+C is ln((|x+1|sqrt(|x^2+1|))/(|x^2+1|))+arctan(x)+C