Question

Evaluate the integrals in Exercises 1–6.∫ dt / (t - √(1 - t²))

Accepted Answer

Start by examining the integral: $$\int \frac{dt}{t - \sqrt{1 - t$$^{2}$$}}. $$Notice the expression in the denominator involves both $$t$$ and $$\sqrt{1 - t$$^{2}$$}$$, which suggests a trigonometric substitution might simplify the square root term. Use the substitution $$t = \sin(	heta)$$, which implies $$dt = \cos(	heta) d	heta. $$This substitution is helpful because $$\sqrt{1 - t$$^{2}$$}$$ becomes $$\sqrt{1 - \sin$$^{2}$$(	heta)} = \cos(	heta). $$Rewrite the integral in terms of $$	heta$$: replace $$t$$ with $$\sin(	heta)$$ and $$dt$$ with $$\cos(	heta) d	heta. $$The integral becomes $$\int \frac{\cos(	heta) d	heta}{\sin(	heta) - \cos(	heta)}. To $$simplify the integral, consider dividing numerator and denominator by $$\cos(	heta) ($$assuming $$\cos(	heta) 
eq 0$$), which transforms the integral into $$\int \frac{d	heta}{	an(	heta) - 1}. $$This form is easier to handle. Next, use the substitution $$u = 	an(	heta) - 1$$, so that $$du = \sec$$^{2}$$(	heta) d	heta. $$Express $$d	heta in $$terms of $$du$$ and rewrite the integral accordingly. This will allow you to integrate with respect to $$u$$ and then back-substitute to $$t.$$

Evaluate the integrals in Exercises 1–6.
∫ dt / (t - √(1 - t²))

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Key Concepts

Integration Techniques

Trigonometric Substitution

Algebraic Manipulation of the Integrand