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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Add a constant to the solution

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

f (x) = \frac{1}{x ^{2}}, (3, \frac{1}{9})

sin^{8} (cos (2 x))

f (x) = sin (x) + x cos (x)

\int 9 sin^{3} (x) cos^{7} (x) d x

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