What is the integral of 1/(sqrt(81+x^2)) ?

The integral of 1/(sqrt(81+x^2)) is ln(1/9 |x+sqrt(81+x^2)|)+C

Popular >

integral of 1/(sqrt(81+x^2))

\int \frac{1}{81 + x ^{2}} d x

ln (\frac{1}{9} x + 81 + x^{2}) + C

Solution steps

\int \frac{1}{81 + x ^{2}} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= \int sec (u) d u

Use the common integral:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{1}{9} x)

= ln tan (arctan (\frac{1}{9} x)) + sec (arctan (\frac{1}{9} x))

Simplify

ln tan (arctan (\frac{1}{9} x)) + sec (arctan (\frac{1}{9} x)) : ln (\frac{1}{9} x + 81 + x^{2})

= ln (\frac{1}{9} x + 81 + x^{2})

Add a constant to the solution

= ln (\frac{1}{9} x + 81 + x^{2}) + C

\frac{\partial}{\partial y} (\frac{2 x}{y ^{3}})

\frac{d y}{d x} = \frac{y ( ln ( 4 y ) - ln ( 4 x ) + 3 )}{3 x}

t^{\frac{9}{2}}

\int - \frac{1}{512} sin (8 x) d x

x \to - 1 lim (\frac{x ^{3} - x}{x - 1})

What is the integral of 1/(sqrt(81+x^2)) ?
The integral of 1/(sqrt(81+x^2)) is ln(1/9 |x+sqrt(81+x^2)|)+C