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Chapter 4: Problem 25

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

Short Answer

Expert verified
Answer: The indefinite integral of the given function is $$\frac{8}{3}x^{\frac{3}{2}} - 8x^{\frac{1}{2}} + C$$, where C is the constant of integration.

Step by step solution

01

Identify the parts to integrate individually

We have a function $$4 \sqrt{x}-\frac{4}{\sqrt{x}}$$ inside the integral. We can rewrite this function as the sum of two simpler functions: $$4x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}$$ We can now integrate each of these functions separately: $$\int\left(4x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\right) dx$$
02

Integrate the first part

To integrate the first part, $$4x^{\frac{1}{2}},$$ we can use the power rule for integration: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ Applying the power rule for n = 1/2: $$\int 4x^{\frac{1}{2}} dx = 4 \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C_1 = \frac{8}{3}x^{\frac{3}{2}} + C_1$$
03

Integrate the second part

Now, we'll integrate the second part, $$-4x^{-\frac{1}{2}},$$ again using the power rule for integration. This time, n = -1/2: $$\int -4x^{-\frac{1}{2}} dx = -4 \frac{x^{-\frac{1}{2} + 1}}{-\frac{1}{2} + 1} + C_2 = -8x^{\frac{1}{2}} + C_2$$
04

Combine the results of the two integrations

Now, we combine the integrals of both parts together: $$\int\left(4x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\right) dx = \frac{8}{3}x^{\frac{3}{2}} - 8x^{\frac{1}{2}} + C$$ Where $$C = C_1 + C_2$$ is the integration constant.
05

Check the result by differentiation

To check if our integration is correct, we will now differentiate the result and make sure it matches the original function inside the integral. Differentiating our result: $$\frac{d}{dx} \left(\frac{8}{3}x^{\frac{3}{2}} - 8x^{\frac{1}{2}} + C\right) = \frac{8}{3} \cdot \frac{3}{2} x^{\frac{1}{2}} - 8 \cdot \frac{1}{2} x^{-\frac{1}{2}} = 4x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}$$ This matches our original given function, so our integration is correct.

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