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

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

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

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

ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣}) + C

Solution steps

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

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}{a} x)

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

Simplify

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

= ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣})

Add a constant to the solution

= ln (\frac{x + a ^{2} + x ^{2}}{∣ a ∣}) + C

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