Question

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.∫ √(x - 2) / √(x - 1) dx

Accepted Answer

Start by examining the integral $$\int \frac{\sqrt{x - 2}}{\sqrt{x - 1}} \, dx. $$Notice the expressions under the square roots are similar, differing by 1. This suggests a substitution to simplify the radicals. Make the substitution $$u = \sqrt{x - 1}. $$Then, express $$x in $$terms of $$u$$: $$x = u^2$ + 1. $$Also, find $$dx in $$terms of $$du$$: differentiate both sides to get $$dx = 2u \, du. $$Rewrite the integral in terms of $$u. $$The numerator becomes $$\sqrt{x - 2} = \sqrt{u^2$ + 1 - 2} = \sqrt{u^2$ - 1}$$, and the denominator is $$\sqrt{x - 1} = u. $$Substitute these and $$dx$$ into the integral to get $$\int \frac{\sqrt{u^2$ - 1}}{u} \cdot 2u \, du. $$Simplify the integral expression: the $$u in $$the denominator and numerator cancel, leaving $$\int 2 \sqrt{u^2$ - 1} \, du. $$Now, to evaluate this integral, use a trigonometric substitution suitable for $$\sqrt{u^2$ - 1}$$, such as $$u = \sec 	heta. $$With $$u = \sec 	heta$$, express $$\sqrt{u^2$ - 1} as$$ $$\sqrt{\sec^2$ 	heta - 1} = 	an 	heta$$, and find $$du = \sec 	heta 	an 	heta \, d	heta. $$Substitute these into the integral and simplify to an integral in terms of $$	heta$$ that can be evaluated using standard trigonometric integrals.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx

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Concetti chiave

Substitution Method in Integration

Trigonometric Substitution

Integrals Involving Radical Expressions