Question

Evaluate the following integrals using the given substitutions.
(a) (3x^2 + 10x)dx/(x^3 + 5x^2 + 18
, substitution u = x3 + 5x2 + 18;
(b)(14x + 4)cos(7x^2 + 4x)dx,substitution u = 7x^2 + 4x.

Answer 1

u = x3 + 5x2 + 18
then
du = 3 x^2 dx + 10 x dx
so in fact we have
u du
integrate u^2/2 + c
so the hard part is squaring
.5 (x^3 + 5 x^2 + 18)^2 + c

Answer 2

(14x + 4)cos(7x^2 + 4x)dx,substitution u = 7x^2 + 4x.

u = 7 x^2 + 4 x
du = 14 x dx + 4 dx = (14x+4)dx
so we have integral of
cos u du
which is

- sin u + c

- sin (7x^2+4x) + c