Section is Integration by Substitution:
Evaluate the indicated integral.
x^2 sec^2 x^3dx
To evaluate the integral ∫x^2 sec^2(x^3) dx, we can use the technique of integration by substitution.
Let's use u = x^3 as our substitution. To find du/dx, we can differentiate both sides with respect to x:
du/dx = 3x^2.
Now, solving for dx, we get dx = du/(3x^2).
Substituting these values into the integral, we have:
∫x^2 sec^2(x^3) dx = ∫(x^2)(sec^2(u))(1/3x^2) du.
Simplifying this, we have:
(1/3) ∫sec^2(u) du.
Now, the integral on the right-hand side is a standard integral that we can solve easily. The integral of sec^2(u) du is equal to tan(u) + C, where C is the constant of integration.
Substituting back u = x^3, we obtain:
(1/3) ∫sec^2(x^3) dx = (1/3) (tan(x^3) + C).
Therefore, the result of the integral is (1/3)(tan(x^3) + C).