Question

The correct evaluation of $\int _ { 0 } ^ { \pi / 2 } \left| \sin \left( x - \frac { \pi } { 4 } \right) \right| d x$ is

• A. $2 + \sqrt { 2 }$
• B. $2 - \sqrt { 2 }$
• C. $- 2 + \sqrt { 2 }$
• D. 0

Answer 1

Answer: (B)

$2 - \sqrt { 2 }$

Answer 2

Let $I = \int _ { 0 } ^ { \pi / 2 } \left| \sin \left( x - \frac { \pi } { 4 } \right) \right| d x$

$x - \frac { \pi } { 4 }$ is –ve when $x \leq \frac { \pi } { 4 }$ and +ve when $x > \frac { \pi } { 4 }$

$= - \int _ { 0 } ^ { \pi / 4 } \sin \left( x - \frac { \pi } { 4 } \right) d x + \int _ { \pi / 4 } ^ { \pi / 2 } \sin \left( x - \frac { \pi } { 4 } \right) d x$ $= 2 - \sqrt { 2 }$ .