Problem 25 - Calculus Early Transcendentals

Question

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(4 \sqrt{x}-\frac{4}{\sqrt{x}}\right) d x$$

Step-by-Step Solution

Answer

Answer: The indefinite integral of the given function is $$\frac{8}{3}x^{3/2}-8x^{1/2}+C$$.

Step 1: Rewrite the expression with x to a power

We rewrite the given expression, in terms of x raised to a power: $$\int\left(4x^{1/2}-\frac{4}{x^{1/2}}\right) d x$$

Step 2: Find the indefinite integral of each term

Now, we find the indefinite integral of each term separately: $$\int 4x^{1/2} dx + \int -\frac{4}{x^{1/2}} dx$$ Using the power rule for indefinite integrals, we get: $$4\int x^{1/2} dx - 4 \int x^{-1/2} dx$$ Apply the power rule, \({\int x^n dx = \frac{x^{n+1}}{n+1}}\), for each term and then combine the results: $$4\frac{x^{1/2+1}}{1/2+1} - 4\frac{x^{-1/2+1}}{-1/2+1} + C$$

Step 3: Simplify the result

Simplify the expression and add the constant of integration, C: $$4\frac{x^{3/2}}{3/2}-4\frac{x^{1/2}}{1/2}+C$$ $$\frac{8}{3}x^{3/2}-8x^{1/2}+C$$

Step 4: Check the result by differentiation

Differentiate the obtained result to check if it matches the original function: $$\frac{d}{dx}\left(\frac{8}{3}x^{3/2}-8x^{1/2}+C\right)$$ Apply the power rule for differentiation, \({\frac{d}{dx}(x^n)=nx^{n-1}}\): $$\frac{8}{3}(3/2)x^{3/2-1}-8(1/2)x^{1/2-1}$$ Simplify the expression: $$4x^{1/2} - \frac{4}{x^{1/2}},$$ which matches the original function in the exercise. The indefinite integral of the given function is: $$\int\left(4 \sqrt{x}-\frac{4}{\sqrt{x}}\right) d x = \frac{8}{3}x^{3/2}-8x^{1/2}+C.$$