7-56. Trigonometric substitutions Evaluate the following integ...

Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

26. ∫[√2 to √2] √(x² - 1)/x dx

Accepted Answer

Evaluate the integral using trigonometric substitution. We have the integral from 22 of 3 divided by 3 to the square root of 2 of 2 square roots of X2 minus 1 divided by X D X. Now to solve this, we first need our substitution. In this case, our substitution will be the form X equals see data. We can then write DX as sequent theta. Multiplied by tangent data, the data. Now, we have X equals 22 of 3, divided by 3. And X equals a square root of 2. We can use this to solve our data. This would give us data equals pi divided by 6. And theta equals pi divided by 4. Let's go ahead and do our substitution. We now have the integral From pi divided by 6 to pi divided by 4. Of 2 square roots. Of the sequent data squared minus 1. Divided by sequent theta multiplied by DX. Which is sin theta Tent data, the data. Let's go ahead and simplify this now. We have the integral, I divided by 6, to pi divided by 4. We seek and the they cancels. We get 2 multiplied. By sequence squared data minus 1, all multiplied by tangent data the data. We also know That change of squared data equals the sequence square minus 1 by our Pythagorean identity. We can then rewrite this as the integral. Of 2 multiplied by the square root. Of tangent squared data multiplied by. Change of data, the data. We can then continue simplifying. We noticed a square root of tangent squared would just give us tangent. We have the integral pi divided by 62 pi divided by 4. I have 2 tangent squared data. Do you Now we can use our previous identity, sequin square the minus 1. This will give us 2 multiplied by the integral, from pi divided by 6 to pi divided by 4. Of secret square theta. -1 Let's take our integral now. We have 2 multiplied by the integral of sequence squared data, which is just tangent data. In the integral of one, which is just data. From pi divided by 6 to pi divided by 4. Now we can plug in our values. We have 2 multiplied. By the tangent, I divided by 4. Minus pi divided by 4. Minus the tangent. Of pi divided by 6 minus pi divided by 6. We can now simplify this. We get 2 multiplied by 1. Minus pi divided by 4. Minus the square root of 3 divided by 3. Plus pi divided by 6. We can now simplify by getting a common denominator. If we get a common denominator, our answer ends up being 2 multiplied. By 12 minus 4 squares of 3. Minus pi divided by 12. We can finally simplify this answer to be Our answer, 12 minus 4 square of 3. Minus pi all divided by 6. This is the final answer to our problem. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx

Key Concepts

Trigonometric Substitution

Integration Techniques

Definite Integrals

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