What is the integral of 1/(sqrt(4x^2+c^2)) ?

The integral of 1/(sqrt(4x^2+c^2)) is 1/2 ln((|2x+sqrt(c^2+4x^2)|)/(|c|))+C

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integral of 1/(sqrt(4x^2+c^2))

\int \frac{1}{4 x ^{2} + c ^{2}} d x

\frac{1}{2} ln (\frac{2 x + c ^{2} + 4 x ^{2}}{∣ c ∣}) + C

Solution steps

\int \frac{1}{4 x ^{2} + c ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{sec ( u )}{2} d u

= \int \frac{sec ( u )}{2} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int sec (u) d u

Use the common integral:

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

= \frac{1}{2} ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{2}{c} x)

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

Simplify

\frac{1}{2} ln tan (arctan (\frac{2}{c} x)) + sec (arctan (\frac{2}{c} x)) : \frac{1}{2} ln (\frac{2 x + c ^{2} + 4 x ^{2}}{∣ c ∣})

= \frac{1}{2} ln (\frac{2 x + c ^{2} + 4 x ^{2}}{∣ c ∣})

Add a constant to the solution

= \frac{1}{2} ln (\frac{2 x + c ^{2} + 4 x ^{2}}{∣ c ∣}) + C

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What is the integral of 1/(sqrt(4x^2+c^2)) ?
The integral of 1/(sqrt(4x^2+c^2)) is 1/2 ln((|2x+sqrt(c^2+4x^2)|)/(|c|))+C