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

The integral of ln(xsqrt(1+x^2)) is 1/2 xln(x^2+1)+xln(x)-2x+arctan(x)+C

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

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

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

Solution steps

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

Simplify

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

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

\int ln (x) d x = x ln (x) - x

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

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

Simplify

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

Add a constant to the solution

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

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