Question

Evaluate the integral

${\int }_{\alpha }^{\pi /2-\alpha }$ $\frac{cotx}{tanx+cotx}dx$

• A. $\frac{\pi }{4}-\alpha$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{8}$

Answer 1

The correct option is A $\frac{\pi }{4}-\alpha$

${\int }_{\alpha }^{\frac{\pi }{2}-\alpha }\frac{cotx}{tanx+cotx}dx$

${\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f\left(\right(a+b)-x)dx$

$I={\int }_{\alpha }^{\frac{\pi }{2}-\alpha }\frac{cotx}{tanx+cotx}dx={\int }_{\alpha }^{\frac{\pi }{2}-\alpha }\frac{tanx}{tanx+cotx}dx=I$

$2I={\int }_{\alpha }^{\frac{\pi }{2}}1\cdot dx$

$I=\frac{(\frac{\pi }{2}-2x)}{2}$

$=\frac{\pi }{4}-\alpha$