Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ dx / (x - √x)

Accepted Answer

Start by rewriting the integral to make it easier to work with. Notice that the denominator is $$x - \sqrt{x}. To $$simplify, let’s express everything in terms of $$\sqrt{x}. $$Set $$t = \sqrt{x}$$, which implies $$x = t^2$ and dx = 2t$$ \, dt$$. Substitute x$$ = $$t^2$$ and $$dx = 2t \, dt$$ into the integral. The integral becomes $$\int \frac{2t \, dt}{t^2$ - t}. $$Simplify the denominator: $$t^2 - t = t(t - 1$$)$$, so the integral is $$\int \frac{2t}{t(t - 1)} \, dt$$. Simplify the integrand by canceling t$ in numerator and denominator, resulting in $$\int \frac{2}{t - 1} \, dt$$. This is a simpler rational function integral. Integrate $$\int \frac{2}{t - 1} \, dt$$ by recognizing it as a standard form $$\int \frac{1}{u} \, du = \ln|u| + C$$. Here, u = t - 1$, so the integral is 2$$ \ln|t - 1| + C$$. Finally, substitute back t$$ = \sqrt{x}$$ to express the answer in terms of x$. The result is 2$$ \ln|\sqrt{x} - 1| + C$$.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ dx / (x - √x)

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Key Concepts

Algebraic Manipulation

Substitution Method

Integration of Rational Functions