What is the integral of ln(x^2+5) ?

The integral of ln(x^2+5) is xln(x^2+5)+2sqrt(5)arctan(x/(sqrt(5)))-2x+C

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integral of ln(x^2+5)

\int ln (x^{2} + 5) d x

x ln (x^{2} + 5) + 25 arctan (\frac{x}{5}) - 2 x + C

Solution steps

\int ln (x^{2} + 5) d x

Apply Integration By Parts

: x ln (x^{2} + 5) - \int \frac{2 x ^{2}}{x ^{2} + 5} d x

= x ln (x^{2} + 5) - \int \frac{2 x ^{2}}{x ^{2} + 5} d x

\int \frac{2 x ^{2}}{x ^{2} + 5} d x = - 25 arctan (\frac{x}{5}) + 2 x

= x ln (x^{2} + 5) - (- 25 arctan (\frac{x}{5}) + 2 x)

Simplify

= x ln (x^{2} + 5) + 25 arctan (\frac{x}{5}) - 2 x

Add a constant to the solution

= x ln (x^{2} + 5) + 25 arctan (\frac{x}{5}) - 2 x + C

f (x) = 4 x^{2} - 4 x + 1, at x = 1

f (x) = (\frac{x + 2}{x - 2})^{2}, at x = 6

f^{^{'}} (x) = \frac{1}{( 1 - x ^{2} ) ^{2}}

\frac{d}{d x} (2 x^{\frac{1}{4}} (x^{3} - 11))

f (x) = 5 x^{2} + 4

What is the integral of ln(x^2+5) ?
The integral of ln(x^2+5) is xln(x^2+5)+2sqrt(5)arctan(x/(sqrt(5)))-2x+C