If-f-x-determinant-sec-2-x-1-1-cos-2-x-cos-2-x-csc-2-x-1-cos-2-x-tan-2-x-evaluate-0-pi-4-f-x-dx-

Question Number 140548 by liberty last updated on 09/May/21

If f (x) = | \begin{matrix} \sec^{2} x 1 1 \\ \cos^{2} x \cos^{2} x \csc^{2} x \\ 1 \cos^{2} x \tan^{2} x \end{matrix} |

evaluate \underset{0}{\int^{π / 4}} f (x) dx .

Answered by EDWIN88 last updated on 09/May/21

I = \underset{0}{\int^{π / 4}} f (x) dx = | \begin{matrix} \underset{0}{\int^{π / 4}} \sec^{2} x dx 1 1 \\ \underset{0}{\int^{π / 4}} \cos^{2} x dx \underset{0}{\int^{π / 4}} \cos^{2} x dx \underset{0}{\int^{π / 4}} \csc^{2} x dx \\ 1 \underset{0}{\int^{π / 4}} \cos^{2} x dx \underset{0}{\int^{π / 4}} \tan^{2} x dx \end{matrix} |

(1) \underset{0}{\int^{π / 4}} \sec^{2} x dx = 1 (2) \underset{0}{\int^{π / 4}} \cos^{2} x dx = \underset{0}{\int^{π / 4}} (\frac{1 + \cos 2 x}{2}) dx = {[\frac{x + \frac{\sin 2 x}{2}}{2}]}_{0}^{π / 4} = \frac{π + 2}{8}

(3) \underset{0}{\int^{π / 4}} \csc^{2} x dx = - 1 (4) \underset{0}{\int^{π / 4}} \tan^{2} x dx = \underset{0}{\int^{π / 4}} (\sec^{2} x - 1) dx = {[\tan x - x]}_{0}^{π / 4} = \frac{4 - π}{4}

I = | \begin{matrix} 1 1 1 \\ \frac{π + 2}{8} \frac{π + 2}{8} - 1 \\ 1 \frac{π + 2}{8} \frac{4 - π}{4} \end{matrix} |

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