Question

Solve the following : `int_0^1 (x^2 + 3x + 2)/sqrt(x)*dx`

Answer 1

Let I = `int_0^1 (x^2 + 3x + 2)/sqrt(x)*dx`

= `int_0^1((x^2 + 3x + 2)/x^(1/2))*dx`

= `int_0^1(x^2/x^(1/2) + (3x)/x^(1/2) + (2)/x^(1/2))*dx`

= `int_0^1(x^(1/2) + 3x^(1/2) + 2x^(1/2))*dx`

= `int_0^1x^(3/2)*dx + 3int_0^1 x^(1/2)*dx + 2int_0^1 x^(1/2)*dx`

= `[(x^5/2)/(5/2)]_0^1 + 3[(x^3/2)/(3/2)]_0^1 + 2[(x^1/2)/(1/2)]_0^1`

= `(2)/(5)(1 - 0)  3 xx (2)/(3)(1 - 0) + 2 xx 2(1 - 0)`

= `(2)/(5)  2 + 4`

∴ I = `(32)/(5)`.