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

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

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

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

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

Solution steps

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

\int \frac{tan ^{4} ( x )}{sec ^{7} ( x )} d x

\int 3 sin^{2} (t) cos (t) d t

\int x^{4} e^{3 x} d x

\frac{d}{d x} (2 sec (2 x) tan (2 x))

0.125 x 64 - x^{2}

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