pls i need help with these Integration problems. Thanks in advance?

Question

int sin^2(4x)cos^2(4x) dx, int 2cos^3x dx limit from 0 to pi/4

Answer 1

∫sin² (4x) cos² (4x) dx

First, let us simplify the expression in this integral using the double-angle formulas:

∫(sin (4x) cos (4x))² dx
∫(sin (8x)/2)² dx
1/4 ∫sin² (8x) dx

Using the formula sin² x = (1-cos (2x))/2:

1/8 ∫1-cos (16x) dx

Now we can integrate easily:

x/8 - sin (16x)/128 + C

And we are done with the first problem. For the second:

[0, π/4]∫2 cos³ x dx

Extract a factor of cos x:

[0, π/4]∫2 cos² x cos x dx

Using the Pythagorean theorem:

[0, π/4]∫2 (1-sin² x) cos x dx

Expanding:

[0, π/4]∫2 cos x - 2 sin² x cos x dx

Integrating (using the inverse chain rule on the expression on the right):

2 sin x - 2/3 sin³ x |[0, π/4]

Evaluating:

2 sin (π/4) - 2/3 sin³ (π/4) - 2 sin 0 + 2/3 sin³ 0
2 √2/2 - 2/3 (√2/2)³
√2 - √2/6
5√2/6

And we are done.