SATHEE: Indefinite Integration 3 Question 17

Indefinite Integration 3 Question 17

17. $f (x) = | \begin{array}{ccc} \sec x & \cos x & \sec^{2} x + \cot x cosec x \cos^{2} x & \cos^{2} x & {cosec}^{2} x 1 & \cos^{2} x & \cos^{2} x \end{array} |$ .

Then, $\int_{0}^{π / 2} f (x) d x = \dots$ .

$(1987, 2 M)$

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Answer:

Correct Answer: 17. $- \frac{15 π + 32}{60}$

Solution:

Given, $f (x) = | \begin{array}{ccc} \sec x & \cos x & cosec x \cdot \cot x + \sec^{2} x \cos^{2} x & \cos^{2} x & {cosec}^{2} x 1 & \cos^{2} x & \cos^{2} x \end{array} |$ Applying $R_{3} \to \frac{1}{\cos x} R_{3}$ ,

$f (x) = \cos x | \begin{array}{ccc} \sec x & \cos x & cosec x \cdot \cot x + \sec^{2} x \cos^{2} x & \cos^{2} x & {cosec}^{2} x \sec x & \cos x & \cos x \end{array} |$

Applying $R_{1} \to R_{1} - R_{3} \Rightarrow f (x)$

$= \cos x | \begin{array}{ccc} 0 & 0 & cosec x \cdot \cot x + \sec^{2} x - \cos x \\ \cos^{2} x & \cos^{2} x & {cosec}^{2} x \\ \sec x & \cos x & \cos x \end{array} |$

$\begin{aligned} = (cosec x \cdot \cot x + \sec^{2} x - \cos x) \cdot (\cos^{3} x - \cos x) \cdot \cos x \\ = - \frac{\sin^{2} x + \cos^{3} x - \cos^{3} x \cdot \sin^{2} x}{\sin^{2} x \cdot \cos^{2} x} \cdot \cos^{2} x \cdot \sin^{2} x \\ = - \sin^{2} x - \cos^{3} x (1 - \sin^{2} x) = - \sin^{2} x - \cos^{5} x \\ ∴ \int_{0}^{π / 2} f (x) d x = - \int_{0}^{π / 2} (\sin^{2} x + \cos^{5} x) d x \\ ∵ \int_{0}^{π / 2} \sin^{m} x \cdot \cos^{n} x d x = \frac{\sqrt{\frac{m + 1}{2}} \sqrt{\frac{n + 2}{2}}}{2 \sqrt{\frac{m + n + 2}{2}}} \\ \int_{0}^{π / 2} f (x) d x = - \frac{\sqrt{\frac{3}{2}} \cdot \sqrt{\frac{1}{2}}}{2 \sqrt{2}} + \frac{\sqrt{\frac{6}{2}} \cdot \sqrt{\frac{1}{2}}}{2 \sqrt{\frac{7}{2}}} \\ = - \frac{1 / 2 \cdot π}{2} + \frac{2 \sqrt{π}}{2 \cdot \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{π}} = - \frac{π}{4} + \frac{8}{15} = - \frac{15 π + 32}{60} \end{aligned}$

Indefinite Integration 3 Question 17

17. $f (x) = | \begin{array}{ccc} \sec x & \cos x & \sec^{2} x + \cot x cosec x \cos^{2} x & \cos^{2} x & {cosec}^{2} x 1 & \cos^{2} x & \cos^{2} x \end{array} |$ .

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Indefinite Integration 3 Question 17

17. f(x)=|sec⁡xcos⁡xsec2⁡x+cot⁡xcosecx cos2⁡xcos2⁡xcosec2x 1cos2⁡xcos2⁡x|.

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17. $f (x) = | \begin{array}{ccc} \sec x & \cos x & \sec^{2} x + \cot x cosec x \cos^{2} x & \cos^{2} x & {cosec}^{2} x 1 & \cos^{2} x & \cos^{2} x \end{array} |$ .