Evaluate (Integral) sin 4x cos^2 4x dx.
A. Cos^3(4x)/3 + C
B. -Cos^3(4x)/3 + C
C. Cos^3(4x)/12 + C
D. -Cos^3(4x)/12 + C
Use the substitution
p=cos(4x)
dp=-4sin(4x)dx
I=∫sin(4x)cos²(4x)dx
=∫(1/4)p²dp
Can you take it from here?
To evaluate the integral ∫(sin(4x)cos^2(4x)) dx, you can use integration by substitution. Here's how:
Step 1: Let u = cos(4x).
Taking the derivative of both sides with respect to x, we get du = -4sin(4x) dx.
Rearranging, we have -1/4 du = sin(4x) dx.
Step 2: Rewrite the integral in terms of u.
The original integral becomes ∫(-1/4)u^2 du.
Step 3: Integrate using the power rule.
∫(-1/4)u^2 du = (-1/4) * (u^3/3) + C.
Step 4: Substitute back for u.
(-1/4) * (u^3/3) + C = (-1/12)u^3 + C.
Step 5: Replace u with cos(4x).
(-1/12) * (cos^3(4x)) + C.
Therefore, the correct answer is option B: -Cos^3(4x)/3 + C.