What is the integral of log_{e}(sqrt(x^2+1)-x) ?

The integral of log_{e}(sqrt(x^2+1)-x) is xln(sqrt(x^2+1)-x)+sqrt(x^2+1)+C

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integral of log_{e}(sqrt(x^2+1)-x)

\int lo g_{e} (x^{2} + 1 - x) d x

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

Solution steps

\int lo g_{e} (x^{2} + 1 - x) d x

Apply Integration By Parts

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

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

\int - \frac{x}{x ^{2} + 1} d x = - x^{2} + 1

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

Simplify

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

Add a constant to the solution

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

3 - 4 x

\int 2 x^{13} e^{x^{7}} d x

x \to 0 lim (\frac{2 ^{x} - 1}{2})

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

\frac{d}{d x} (8 x ln (x))

What is the integral of log_{e}(sqrt(x^2+1)-x) ?
The integral of log_{e}(sqrt(x^2+1)-x) is xln(sqrt(x^2+1)-x)+sqrt(x^2+1)+C