What is the integral of 1/(sqrt(9+y^2)) ?

The integral of 1/(sqrt(9+y^2)) is ln(1/3 |y+sqrt(9+y^2)|)+C

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integral of 1/(sqrt(9+y^2))

\int \frac{1}{9 + y ^{2}} d y

ln (\frac{1}{3} y + 9 + y^{2}) + C

Solution steps

\int \frac{1}{9 + y ^{2}} d y

Apply Trigonometric Substitution

= \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}{3} y)

= ln tan (arctan (\frac{1}{3} y)) + sec (arctan (\frac{1}{3} y))

Simplify

ln tan (arctan (\frac{1}{3} y)) + sec (arctan (\frac{1}{3} y)) : ln (\frac{1}{3} y + 9 + y^{2})

= ln (\frac{1}{3} y + 9 + y^{2})

Add a constant to the solution

= ln (\frac{1}{3} y + 9 + y^{2}) + C

What is the integral of 1/(sqrt(9+y^2)) ?
The integral of 1/(sqrt(9+y^2)) is ln(1/3 |y+sqrt(9+y^2)|)+C