Question

Evaluate : $\int _ { \pi / 4 } ^ { \pi / 3 } x$ cosec²x dx –

• A. $\frac { \pi } { 12 \sqrt { 3 } }$(3 $\sqrt { 3 }$ – 4) + $\frac { 1 } { 2 }$ln $\left( \frac { 3 } { 2 } \right)$
• B. $\frac { \pi } { 12 \sqrt { 3 } }$(3 $\sqrt { 3 }$ – 4) + $\left( \frac { 2 } { 3 } \right)$
• C. $\frac { \pi } { 12 \sqrt { 3 } }$(3 $\sqrt { 3 }$ – 2) + $\frac { 1 } { 2 }$ln $\left( \frac { 3 } { 2 } \right)$
• D. None of these

Answer 1

Answer: (A)

$\frac { \pi } { 12 \sqrt { 3 } }$(3 $\sqrt { 3 }$ – 4) + $\frac { 1 } { 2 }$ln $\left( \frac { 3 } { 2 } \right)$

Answer 2

Let I = cosec² x dx

Integrating by parts taking x as first and cosec² x as second function, we have

I = x (– cot x) +

= –$\left( \frac { \pi } { 3 } \cdot \frac { 1 } { \sqrt { 3 } } - \frac { \pi } { 4 } \right)$ + ln | sin x |

= –$\frac { \pi } { 4 }$ + $\left( \ln \frac { \sqrt { 3 } } { 2 } - \ln \frac { 1 } { \sqrt { 2 } } \right)$

= $\frac { \pi } { 12 \sqrt { 3 } }$ (3 $\sqrt { 3 }$ – 4) + $\left( \frac { 3 } { 2 } \right)$.