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

The integral of ln(1+sqrt(1-x^2)) is xln(1+sqrt(1-x^2))-(-arcsin(x)+x)+C

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integral of ln(1+sqrt(1-x^2))

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

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

Solution steps

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

Apply Trigonometric Substitution

: \int ln (1 + cos (u)) cos (u) d u

= \int ln (1 + cos (u)) cos (u) d u

Apply Integration By Parts

: ln (1 + cos (u)) sin (u) - \int - 1 + cos (u) d u

= ln (1 + cos (u)) sin (u) - \int - 1 + cos (u) d u

\int - 1 + cos (u) d u = - u + sin (u)

= ln (1 + cos (u)) sin (u) - (- u + sin (u))

Substitute back

u = arcsin (x)

= ln (1 + cos (arcsin (x))) sin (arcsin (x)) - (- arcsin (x) + sin (arcsin (x)))

Simplify

ln (1 + cos (arcsin (x))) sin (arcsin (x)) - (- arcsin (x) + sin (arcsin (x))) : x ln (1 + 1 - x^{2}) - (- arcsin (x) + x)

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

Add a constant to the solution

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

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What is the integral of ln(1+sqrt(1-x^2)) ?
The integral of ln(1+sqrt(1-x^2)) is xln(1+sqrt(1-x^2))-(-arcsin(x)+x)+C