`int_1^4 1/(xsqrt(16x^2-5)) dx` Evaluate the definite integral

Make the substitution `u = sqrt(16x^2 - 5),` then `16x^2 = u^2 + 5,`

`du = (32x)/(2sqrt(16x^2 - 5)) dx = (16 x dx)/sqrt(16x^2 - 5),`

or `dx/sqrt(16x^2 - 5) = (du)/(16 x).`

The limits of integration for `u` are from `sqrt(11)` to `sqrt(251).`

The indefinite integral becomes `int (du)/(16x^2) = int (du)/(u^2 + 5),`

which is equal to `1/sqrt(5) arctan(u/sqrt(5)) + C.`

This way the definite integral is

`1/sqrt(5) (arctan(sqrt(251/5)) - arctan(sqrt(11/5))) approx0.2026.`

Blog