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

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

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

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

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

Solution steps

\int \frac{1}{2 + 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}{2} x)

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

Simplify

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

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

Add a constant to the solution

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

\int \frac{1}{x ^{2}} e^{l n (x)} d x

\frac{d}{d x} (\frac{2 x ^{3}}{3} - 8 x - 5)

y = tan (7 x), y = 2 sin (7 x)

x \frac{d y}{d x} + y = y^{2}

x^{2} + x

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