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Evaluate the integral.$$\int_{\sqrt{2}}^{2} \frac{\sqrt{2 x^{2}-4}}{x} d x$$
$2-\frac{\pi}{2}$
Calculus 1 / AB
Calculus 2 / BC
Chapter 7
PRINCIPLES OF INTEGRAL EVALUATION
Section 4
Trigonometric Substitutions
Integrals
Integration
Integration Techniques
Trig Integrals
Trig Substitution
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Boston College
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We have given the integral dinero to rotunda 2 spins 4, divided by x into d s. Now substitute here x is equal to root 2 sec theta. When we substitute x equal to root 2 sec theta here, then you can remove the square root from here. So take the derivative both sides, so d x, is equal to root 2. As you know, the derivative of sex theta. We can write x 50 into tan theta into d theta, now substitute the value of x and d x, given integral first here compute. The limit in terms of theta when x is equal to root 2. The value of theta is, we can write when x is also rotolo. Theta is so we can write here 0. When x is equal to 2, then the value of theta is pi by 4, now substitute here so 2 into root, 2 sec fictile whole square. Minus 4 divided by root 2 sec theta and the place of dean, write root 2 sec theta into tan theta into de tita. Now simplify this expression. Then we will get from here, then we'll get from here 2 into 0 to pi by 4 tan theta into tan theta into d theta. Now we can write like that 2 into 025 by 4, here 10 square theta into detent. As you know, the integration of the 10 square theta is equal to sec theta. Sorry. Here we can write in the place of 10 square theta sec square theta, minus 1 point. As you know, 10 square theta is equal to sec square theta minus 1 into the theta now integrate integrate here. The integration of the sac square theta is equal to tan theta, and the integration of the 1 is theta into 2 ketland. The limit to pi by 4 point now plug in the limit here so first we'll plug upper limit, so 210 pi by 4 minus 2 into pi by 4. When you plug here 0, then we'll get all terms are 0 here now the value of 10 pi by 4 is 12 into 1 minus pi by 2. So the answer of this question is 2 minus pi by 2 now simplify more then we can write 4 minus pi by 2. So this is the answer of the given integral. Thank you.
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