How do you integrate [(x^2)(cos(2(x^3)))]? I tried to integrate by parts but I'm going in circles yet again...
Let w = 2x^3
6x^2 dx = dw
The integral becomes the integral of
(1/6)w cos w, which can be solved using integration by parts to give
(1/6)[cos w + w sin w]
= (1/6)[cos(2x^3) + 2x^2 sin(2x^3)]
Wouldn't it simply be
(1/6)sin(2x^3) + c ?
The derivative of Reiny's answer is
(1/6)cos(2x^3)*6x^2 = x^2 cos(2x^3), so I must have made a mistake somewhere.
To integrate the given expression, [(x^2)(cos(2(x^3))], you can use the technique of substitution. Here is a step-by-step explanation of how to do it:
1. Start by letting u = 2(x^3)
This choice of u allows us to simplify the expression and make it easier to integrate.
2. Find du/dx (the derivative of u) and solve it for dx.
In this case, du/dx = 6(x^2),
So, dx = (1/6(x^2)) du
3. Substitute u and dx into the integral.
The integral becomes:
∫ [(x^2)(cos(2(x^3))] dx
= ∫ [(x^2)(cos(u))] ((1/6(x^2)) du)
= (1/6) ∫ cos(u) du
4. Integrate the simplified expression.
The integral of cos(u) is sin(u).
Therefore, the final result is:
(1/6) ∫ cos(u) du = (1/6) sin(u)
5. Replace u with its original expression.
Since u = 2(x^3), substitute it back into the equation:
(1/6) sin(u) = (1/6) sin(2(x^3))
6. Simplify the result, if needed.
The integral is now evaluated. You can leave the answer as (1/6) sin(2(x^3)) or simplify it further as needed.
Note: Sometimes, integration requires multiple attempts before finding the correct approach. If you're stuck, it's important to be patient and try different methods like substitution, parts, or other integration techniques.