Question

The correct evaluation of $\int _ { 0 } ^ { \pi / 2 } \sin x \sin 2 x$ is

• A. $\frac { 4 } { 3 }$
• B. $\frac { 1 } { 3 }$
• C. $\frac { 3 } { 4 }$
• D. $\frac { 2 } { 3 }$

Answer 1

Answer: (D)

$\frac { 2 } { 3 }$

Answer 2

Let $I = \int _ { 0 } ^ { \pi / 2 } \sin x \sin 2 x d x = 2 \int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 } x \cos x d x$

Put $t = \sin x \Rightarrow d t = \cos x d x$

Now, $I = 2 \int _ { 0 } ^ { 1 } t ^ { 2 } d t = \frac { 2 } { 3 } \left[ t ^ { 3 } \right] _ { 0 } ^ { 1 } = \frac { 2 } { 3 }$.