What is the integral of ln(x^2+1)+6 ?

The integral of ln(x^2+1)+6 is xln(x^2+1)+2arctan(x)+4x+C

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integral of ln(x^2+1)+6

integral

ln (x^{2} + 1) + 6

x ln (x^{2} + 1) + 2 arctan (x) + 4 x + C

Solution steps

\int ln (x^{2} + 1) + 6 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 ln (x^{2} + 1) d x + \int 6 d x

\int ln (x^{2} + 1) d x = x ln (x^{2} + 1) - 2 (- arctan (x) + x)

\int 6 d x = 6 x

= x ln (x^{2} + 1) - 2 (- arctan (x) + x) + 6 x

Simplify

x ln (x^{2} + 1) - 2 (- arctan (x) + x) + 6 x : x ln (x^{2} + 1) + 2 arctan (x) + 4 x

= x ln (x^{2} + 1) + 2 arctan (x) + 4 x

Add a constant to the solution

= x ln (x^{2} + 1) + 2 arctan (x) + 4 x + C

\frac{1}{( t ^{2} + t )}

\frac{( 2 x + 1 )}{( x + 2 )}

\frac{1}{x ( x ^{8} - 1 )}

\frac{( 8 x + 2 )}{x}

\frac{5}{x ^{4}}

What is the integral of ln(x^2+1)+6 ?
The integral of ln(x^2+1)+6 is xln(x^2+1)+2arctan(x)+4x+C