Find the value of the integral $\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x$
A) $\frac { \sqrt { 2 } } { 18 }$

B) $\frac { \pi } { 4 }$
C) $\frac { 3 \pi } { 2 }$
D) $\frac { \sqrt { 2 } } { 12 }$
E) $\frac { \sqrt { 2 } } { 6 }$
F) $\frac { \sqrt { 2 } } { 9 }$

G) $\frac { 2 \pi } { 3 }$

H) $\frac { \pi } { 2 }$

Question

Find the value of the integral $\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x$ 
A) $\frac { \sqrt { 2 } } { 18 }$

B) $\frac { \pi } { 4 }$ 
C) $\frac { 3 \pi } { 2 }$ 
D) $\frac { \sqrt { 2 } } { 12 }$ 
E) $\frac { \sqrt { 2 } } { 6 }$
F) $\frac { \sqrt { 2 } } { 9 }$

G) $\frac { 2 \pi } { 3 }$

H) $\frac { \pi } { 2 }$

Quizplus · Accepted Answer

Using the substitution $u = \sin x$, we have $du = \cos x dx$. Substituting into the integral, we get$$\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x = \int _ { 0 } ^ { \sin (x/4) } u ^ { 2 } d u = \left [ \frac { u ^ { 3 } } { 3 } \right ] _ { 0 } ^ { \sin (x/4) } = \frac { \sin ^ { 3 } (x/4) } { 3 }$$Therefore, the value of the integral is $\frac { \sqrt { 2 } } { 12 }$.

Find the Value of the Integral ∫0x/4sin⁡2xcos⁡xdx\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x∫0x/4​sin2xcosxdx

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