What is the integral of 1/(sqrt(x^2+5)) ?

The integral of 1/(sqrt(x^2+5)) is ln| 1/(sqrt(5))x+sqrt(1/5 (5+x^2))|+C

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integral of 1/(sqrt(x^2+5))

\int \frac{1}{x ^{2} + 5} d x

ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}) + C

Solution steps

\int \frac{1}{x ^{2} + 5} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= \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}{5} x)

= ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))

Simplify

ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) : ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})

= ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2})

Add a constant to the solution

= ln \frac{1}{5} x + \frac{1}{5} (5 + x^{2}) + C

sin (y) d x + cos (x) d y = 0

x, - 3, 16, 1

\int y^{2} z + 2 x z^{3} d x

\frac{d}{d x} (2 (16 e^{4 x} x + 8 e^{4 x}))

\int \frac{3}{x ^{2} - 6 x - 7} d x

What is the integral of 1/(sqrt(x^2+5)) ?
The integral of 1/(sqrt(x^2+5)) is ln| 1/(sqrt(5))x+sqrt(1/5 (5+x^2))|+C