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

The integral of a/(sqrt(u^2+a^2)) is aln((|u+sqrt(a^2+u^2)|)/(|a|))+C

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integral of a/(sqrt(u^2+a^2))

\int \frac{a}{u ^{2} + a ^{2}} d u

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

Solution steps

\int \frac{a}{u ^{2} + a ^{2}} d u

Take the constant out:

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

= a \cdot \int \frac{1}{u ^{2} + a ^{2}} d u

Apply Trigonometric Substitution

: \int sec (v) d v

= a \cdot \int sec (v) d v

Use the common integral:

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

= a ln ∣ tan (v) + sec (v) ∣

Substitute back

v = arctan (\frac{1}{a} u)

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of a/(sqrt(u^2+a^2)) ?
The integral of a/(sqrt(u^2+a^2)) is aln((|u+sqrt(a^2+u^2)|)/(|a|))+C