Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$
A)1
B) $\frac { \pi } { 16 }$
C) $\frac { \pi } { 4 }$
D)2
E) $\frac { \pi } { 32 }$
F) $\frac { \pi } { 8 }$
G) $\frac { \pi } { 64 }$
H) $\frac { 1 } { 6 }$

Question

Find the value of the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$ 
A)1
B) $\frac { \pi } { 16 }$ 
C) $\frac { \pi } { 4 }$ 
D)2
E) $\frac { \pi } { 32 }$ 
F) $\frac { \pi } { 8 }$ 
G) $\frac { \pi } { 64 }$ 
H) $\frac { 1 } { 6 }$

Quizplus · Accepted Answer

To solve the integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$, we can use the double angle identity for sine and cosine, which is $\sin^2(x)\cos^2(x) = \frac{1}{4}\sin^2(2x)$. However, the correct approach here involves recognizing that $\sin^2(2x)\cos^2(2x) = (\frac{1}{2}\sin(4x))^2$, and then applying the power reduction formula, or directly using the identity for $\sin^2(x)\cos^2(x) = \frac{1}{4}(1 - \cos(4x))$. Integrating this from $0$ to $\pi/4$ gives $\frac{\pi}{32}$.

Find the Value of the Integral ∫0π/4sin⁡2(2x)cos⁡2(2x)dx\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x∫0π/4​sin2(2x)cos2(2x)dx

Find the Value of the Integral $\int _ { 0 } ^ { \pi / 4 } \sin ^ { 2 } ( 2 x ) \cos ^ { 2 } ( 2 x ) d x$