Using an integration formula,what is the indefinite integral of (sign for integral)(cos(4x)+2x^2)(sin(4x)-x)dx. Any help very much appreciated.
An interesting problem! In fact, it would make a good exam question, because you need to know four rules to calculate the integral, or equivalent to four short questions.
(cos(4x)+2x^2)(sin(4x)-x)
=cos(4x)sin(4x)-xcos(4x)+2x²sin(4x)-2x³
1. cos(4x)sin(4x)
use substitution
u=sin(4x)
du=4cos(4x)dx
so
∫cos(4x)sin(4x)dx
=(1/4)∫udu (use the power rule)
2. -xcos(4x)
Use integration by parts:
∫-xcos(4x)dx
=-(1/4)xsin(4x)+(1/4)∫sin(4x)
3. 2x²sin(4x)
Use integration by parts, similar to (2) above.
4. -2x³
use the power rule.
Post your answer for a check if you wish. Do not forget the constant of integration (C).
To find the indefinite integral of the expression (cos(4x) + 2x^2)(sin(4x) - x), you can use the technique of integration by parts.
Integration by parts is based on the product rule of differentiation, which states that d(uv) = u dv + v du, where u and v are functions. By rearranging this equation, we get ∫u dv = uv - ∫v du.
In this case, we can assign u = cos(4x) + 2x^2 and dv = (sin(4x) - x) dx. Let's find du and v:
du = d(cos(4x) + 2x^2) = -4 sin(4x) dx + 4x dx
v = ∫(sin(4x) - x) dx
Now, let's compute v:
To integrate sin(4x), we can use a substitution. Let w = 4x, then dw = 4 dx.
∫sin(4x) dx = ∫sin(w) (1/4) dw = (-1/4) cos(w) + C = (-1/4) cos(4x) + C
To integrate -x, we simply apply the power rule for integration:
∫-x dx = (-1/2)x^2 + C
Now, let's substitute these values back into the integration by parts formula:
∫u dv = uv - ∫v du
∫(cos(4x) + 2x^2)(sin(4x) - x) dx = [(cos(4x) + 2x^2)((-1/4) cos(4x))] - ∫((-1/4) cos(4x)) (-4 sin(4x) dx + 4x dx)
Simplifying, we get:
= (-1/4)cos^2(4x) - (1/2)x^2cos(4x) - (1/4)∫4x sin(4x) dx + ∫x cos(4x) dx
Now, we can evaluate the remaining integrals. To integrate 4x sin(4x) dx, we can use integration by parts once again, applying the same method outlined above.
By following these steps, you can calculate the indefinite integral of the expression (cos(4x) + 2x^2)(sin(4x) - x).