Integral 1squareroot 4 x2 SolutionAnswer 14x2 dx Substitut

Integral 1/squareroot 4 + x^2 =

Solution

Answer.

1/(4+x²) dx

Substitute

u = x/2

du/dx = 1/2

then,

= 1/(u²+1) du

We make a trigonometric substitution now:

u = tan(v)

v = arctan(u),

du/dv=sec²(v)

so,

= sec²(v)/(tan²(v)+1) dv

Simplify (tan²(v)+1) = sec²(v):

= sec(v) dv

= ln(tan(v)+sec(v))

Undo substitution v = arctan(u):

=ln((u²+1) + u)

Undo substitution u = x/2:

= ln((x²/4 + 1) + x²)

Solution is:

= ln ((4 + x²) + x) + C