Starting with F(x) = integral (-2, x) 3t^2(cos (t^3) + 2) dt, use substitution u(t)=t^3 to rewrite the definite integral. You should get a new equivalent expression for F(x), which consists of this new definite integral.
u = t^3
so t = u^(1/3)
du = 3 t^2 dt
so dt = du/3 t^2 = du /3u^(2/3)
if t = -2 , u = -8
if t = x, u = x^3
To rewrite the definite integral using the substitution u(t) = t^3, we need to find the corresponding limits of integration in terms of u.
First, let's find the new lower limit of integration:
When t = -2, we have u = (-2)^3 = -8.
Next, let's find the new upper limit of integration:
When t = x, we have u = x^3.
Therefore, the new definite integral using the substitution u(t) = t^3 is:
∫[u(-8), u(x)] 3(u^(2/3))(cos(u) + 2) du
So, the new equivalent expression for F(x) is:
F(x) = ∫[u(-8), u(x)] 3(u^(2/3))(cos(u) + 2) du
To rewrite the definite integral using the substitution u(t) = t^3, we first need to substitute t = u^(1/3) into the existing integral expression.
Given: F(x) = integral (-2, x) 3t^2(cos(t^3) + 2) dt
Using the substitution u(t) = t^3, we get:
F(x) = integral (-2, x) 3(u^(1/3))^2(cos(u) + 2) (du/dt) dt
Next, we need to find du/dt, which is the derivative of u with respect to t.
Taking the derivative of both sides of the substitution equation u(t) = t^3, we get:
du/dt = 3t^2
Now, substituting du/dt = 3t^2 back into the integral expression, we have:
F(x) = integral (-2, x) 3(u^(1/3))^2(cos(u) + 2) (3t^2) dt
Now, simplify the expression further:
F(x) = 9 integral (-2, x) t^2(u^(2/3))(cos(u) + 2) dt
Alternatively, we can rewrite the u(t) substitution limits:
When t = -2, u = (-2)^3 = -8
When t = x, u = x^3
Therefore, we can rewrite the integral in terms of u as:
F(x) = 9 integral (-8, x^3) t^2(u^(2/3))(cos(u) + 2) dt
This new expression for F(x) consists of the new definite integral that includes the substitution u(t) = t^3.