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.
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
(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
To evaluate the given integrals using the given substitutions, we'll follow the steps of integration by substitution.
(a) ∫((3x^2 + 10x)dx)/(x^3 + 5x^2 + 18), substitution u = x^3 + 5x^2 + 18:
1. Differentiate u with respect to x to find du/dx:
du/dx = (3x^2 + 10x)
2. Rearrange the differential equation to solve for dx:
du = (3x^2 + 10x)dx
3. Substitute the value of dx with du from the above equation in the original integral:
∫(du)/(u)
4. The integral becomes ∫du/u, which is a known integral:
∫(du)/(u) = ln|u| + C
5. Substitute back the value of u:
ln|x^3 + 5x^2 + 18| + C
Therefore, the solution to the integral is ln|x^3 + 5x^2 + 18| + C.
(b) ∫((14x + 4)cos(7x^2 + 4x)dx), substitution u = 7x^2 + 4x:
1. Differentiate u with respect to x to find du/dx:
du/dx = (14x + 4)
2. Rearrange the differential equation to solve for dx:
dx = (du)/(14x + 4)
3. Substitute the value of dx with du/(14x + 4) in the original integral:
∫((14x + 4)cos(7x^2 + 4x))(du)/(14x + 4)
4. Simplify the integral:
∫(cos(u))du
5. The integral of cos(u) is sin(u):
∫(cos(u))du = sin(u) + C
6. Substitute back the value of u:
sin(7x^2 + 4x) + C
Therefore, the solution to the integral is sin(7x^2 + 4x) + C.