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

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

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integral of 2/(sqrt(9+x^2))

\int \frac{2}{9 + x ^{2}} d x

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

Solution steps

\int \frac{2}{9 + x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{1}{9 + x ^{2}} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= 2 \cdot \int sec (u) d u

Use the common integral:

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

= 2 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{1}{3} x)

= 2 ln tan (arctan (\frac{1}{3} x)) + sec (arctan (\frac{1}{3} x))

Simplify

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

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

Add a constant to the solution

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

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