l i m x → π − 2 ∫ x 3 ( π 2 ) 2 c o s t 1 / 3 d t ( x − π 2 ) 2

l i m h → 0 ∫ ( π 2 − h ) ( π 2 ) 3 c o s ( t 1 / 3 ) d t h 2 = l i m h → 0 0 + 3 ( π 2 − h ) 2 c o s ( π 2 − h ) 2 h = l i m h → 0 3 ( π 2 − h ) 2 s i n h 2 h = 3 π 2 8

$\underset{x \to \frac{π^{-}}{2}}{l i m} \frac{\int_{x^{3}}^{{(\frac{π}{2})}^{2}} c o s t^{1 / 3} d t}{{(x - \frac{π}{2})}^{2}}$

Option 1 - <math> <mrow> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math>

Option 2 - <math> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>4</mn> </mrow> </mfrac> </mrow> </math>

Option 3 - <math> <mrow> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mi>π</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> </mrow> </mfrac> </mrow> </math>

Option 4 - <math> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mrow> <mn>8</mn> </mrow> </mfrac> </mrow> </math>

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Answered by

Alok Kumar Singh | Contributor-Level 10

4 months ago

Correct Option - 3

Detailed Solution:

$\underset{h \to 0}{l i m} \frac{\int_{(\frac{π}{2} - h)}^{{(\frac{π}{2})}^{3}} c o s (t^{1 / 3}) d t}{h^{2}}$

$= \underset{h \to 0}{l i m} \frac{0 + 3 {(\frac{π}{2} - h)}^{2} c o s (\frac{π}{2} - h)}{2 h}$

$= \underset{h \to 0}{l i m} \frac{3 {(\frac{π}{2} - h)}^{2} s i n h}{2 h}$

$= \frac{3 π^{2}}{8}$

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$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{8 \sqrt[]{2} c o s x}{(1 + e^{s i n x}) (1 + s i n^{4} x)}$ dx = ap + b log $(3 + 2 \sqrt[]{2})$

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$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{8 \sqrt[]{2} c o s x}{(1 + e^{s i n x}) (1 + s i n^{4} x)}$ dx

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