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  • Evaluate the definite integrals in Exercises 1 to 20. 8 from pi/6 to pi/4 cosecx dx

Evaluate the definite integrals in Exercises 1 to 20.

    Q8.    \int_\frac{\pi}{6}^\frac{\pi}{4}\textup{cosec}xdx

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Given integral: \int_\frac{\pi}{6}^\frac{\pi}{4}\textup{cosec}xdx

Consider the integral \int\textup{cosec}xdx

\int\textup{cosec}xdx = \log|cosec x -\cot x |

So, we have the function of x, f(x) =\log|cosec x -\cot x |

Now, by Second fundamental theorem of calculus, we have

I = f(\frac{\pi}{4}) -f(\frac{\pi}{6})

= \log|cosec \frac{\pi}{4} -\cot \frac{\pi}{4} | - \log|cosec \frac{\pi}{6} -\cot \frac{\pi}{6} |

= \log|\sqrt2 -1 | - \log|2 -\sqrt3 |

= \log \left ( \frac{\sqrt2 -1}{2-\sqrt3} \right )

 

Posted by

Divya Prakash Singh

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