Evaluate the integral - 2 3 - - 2x- - 5 4x-5 - 2x- - 5 4x-5 - dx

The correct option is A 7 √(7) 2√(5)3√(2) Let I=∫ _ 2 ^ 3 [ √( 2x )√( 5( 4x 5 ) #160;) +√( 2x+ )√( 5( 4x 5 ) #160;) #160;] dx Put 4x 5= ...

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Byju's Answer

Standard XII

Mathematics

Integration of Piecewise Continuous Functions

Evaluate the ...

Question

Evaluate the integral $\int_{2}^{3} [\sqrt{2 x -} \sqrt{5 (4 x - 5)} + \sqrt{2 x +} \sqrt{5 (4 x - 5)}] d x$

\frac{7 \sqrt{7} - 2 \sqrt{5}}{3 \sqrt{2}}

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\frac{7 \sqrt{7} - 4 \sqrt{5}}{3 \sqrt{2}}

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\frac{7 \sqrt{6} - 2 \sqrt{5}}{3 \sqrt{2}}

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\frac{7 \sqrt{7} - 2 \sqrt{5}}{3 \sqrt{5}}

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Solution

The correct option is A $\frac{7 \sqrt{7} - 2 \sqrt{5}}{3 \sqrt{2}}$
Let $I = \int_{2}^{3} [\sqrt{2 x -} \sqrt{5 (4 x - 5)} + \sqrt{2 x +} \sqrt{5 (4 x - 5)}] d x$
Put $4 x - 5 = 5 t^{2} \Rightarrow 4 d x = 10 t d t$
$I = \frac{5}{2} \int_{\sqrt{\frac{3}{5}}}^{\sqrt{\frac{7}{5}}} (\sqrt{\frac{5}{2} ({1 + t}^{2}) - 5 t} + \sqrt{\frac{5}{2} ({1 + t}^{2}) + 5 t}) t d t$
$= {(\frac{5}{2})}^{\frac{3}{2}} \int_{\sqrt{\frac{3}{5}}}^{\sqrt{\frac{7}{5}}} (| t - 1 | + | t + 1 |) t d t .$
$= {(\frac{5}{2})}^{\frac{3}{2}} ⎡ ⎣ \int_{\sqrt{\frac{3}{5}}}^{1} (1 - t) + | (1 + t) | t d t + \int_{1}^{\sqrt{\frac{7}{5}}} ((t - 1) + (t + 1)) t d t ⎤ ⎦$
$= {(\frac{5}{2})}^{\frac{3}{2}} ⎡ ⎣ 2 \int_{\sqrt{\frac{3}{5}}}^{1} t d t + \int_{1}^{\sqrt{\frac{7}{5}}} t^{2} d t ⎤ ⎦ = \frac{7 \sqrt{7} - 2 \sqrt{5}}{3 \sqrt{2}} .$

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Question 6

Which of the following expressions show that rational numbers are associative under multiplication.
(a) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = (\frac{2}{3} \times \frac{- 6}{7}) \times \frac{3}{5}$
(b) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = \frac{2}{3} \times (\frac{3}{5} \times \frac{- 6}{7})$
(c) $\frac{2}{3} \times (\frac{- 6}{7} \times \frac{3}{5}) = (\frac{3}{5} \times \frac{2}{3}) \times \frac{- 6}{7}$
(d) $(\frac{2}{3} \times \frac{- 6}{7}) \times \frac{3}{5} = (\frac{- 6}{7} \times \frac{2}{3}) \times \frac{3}{5}$

Question 6

2/3 x [-6/7+4/5]=[ 2/3 x 4/5] x (-6/7)

Q. Question 11
Find the values of a and b in each of the following

(i) \frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}} = a - 6 \sqrt{3}

(i i) \frac{3 - \sqrt{5}}{3 + 2 \sqrt{5}} = a \sqrt{5} - \frac{19}{11}

(i i i) \frac{\sqrt{2} + \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}} = 2 - b \sqrt{6}

(i v) \frac{7 + \sqrt{5}}{7 - \sqrt{5}} - \frac{7 - \sqrt{5}}{7 + \sqrt{5}} = a + \frac{7}{11} \sqrt{5} b

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Evaluate the integral ∫32[√2x−√5(4x−5)+√2x+√5(4x−5)]dx

Evaluate the integral $\int_{2}^{3} [\sqrt{2 x -} \sqrt{5 (4 x - 5)} + \sqrt{2 x +} \sqrt{5 (4 x - 5)}] d x$