Question

Evaluate the integrals in Exercises 39–56.55. ∫dx/(2√x + 2x)

Accepted Answer

Start by rewriting the integral to make it easier to work with. The integral is given as $$\int \frac{dx}{2\sqrt{x} + 2x}. $$Factor out the common factor in the denominator: $$2(\sqrt{x} + x)$$, so the integral becomes $$\int \frac{dx}{2(\sqrt{x} + x)}. $$Simplify the integral by factoring out the constant $$\frac{1}{2}$$: $$\frac{1}{2} \int \frac{dx}{\sqrt{x} + x}. $$This makes the integral $$\frac{1}{2} \int \frac{dx}{\sqrt{x} + x}. To $$handle the terms involving $$\sqrt{x}$$ and $$x$$, use the substitution $$t = \sqrt{x}. $$Then, $$x = t^2$ and dx = 2t$$ \, dt$$. Substitute these into the integral to rewrite it in terms of t$. After substitution, the integral becomes $$\frac{1}{2} \int \frac{2t \, dt}{t + $$t^2$$}$$. Simplify the numerator and denominator inside the integral to get $$\int \frac{t}{t + $$t^2$$} \, dt$$. Simplify the integrand by factoring t$ in the denominator: t$$ + $$t^2 = t(1 + t$$)$$. Then the integrand becomes $$\frac{t}{t(1 + t)} = \frac{1}{1 + t}$$. Now, the integral reduces to $$\int \frac{1}{1 + t} \, dt$$, which is a standard integral.

Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)

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Key Concepts

Integration of Rational Functions

Substitution Method

Simplifying Expressions with Radicals