] Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.)

Ask by Ortega Guerrero. in the United States

Mar 28,2025

Question

] Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.) $$ \int \frac{4 d x}{3 \sqrt{x^{3}}} $$

UpStudy AI · Accepted Answer

We start with the given integral

$$
\int \frac{4 \, dx}{3 \sqrt{x^3}}.
$$

### Step 1. Rewrite the Integral

Express the square root in exponential form. Since

$$
\sqrt{x^3} = x^{3/2},
$$

the integral becomes

$$
\int \frac{4 \, dx}{3x^{3/2}} = \frac{4}{3} \int x^{-3/2} \, dx.
$$

### Step 2. Evaluate the Integral

Use the power rule for integration. Recall that for $$ n \neq -1 $$,

$$
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C.
$$

Here, $$ n = -\frac{3}{2} $$, so

$$
n+1 = -\frac{3}{2} + 1 = -\frac{1}{2}.
$$

Thus,

$$
\int x^{-3/2} \, dx = \frac{x^{-1/2}}{-1/2} + C = -2x^{-1/2} + C.
$$

Now, multiplying by the constant factor $$\frac{4}{3}$$, we get

$$
\frac{4}{3} \int x^{-3/2} \, dx = \frac{4}{3} \left(-2x^{-1/2}\right) + C = -\frac{8}{3} x^{-1/2} + C.
$$

### Step 3. Final Answer

The evaluated integral is

$$
-\frac{8}{3} x^{-1/2} + C,
$$

or equivalently,

$$
-\frac{8}{3 \sqrt{x}} + C.
$$

### Step 4. Verification by Differentiating

Differentiate the result with respect to $$x$$. Let

$$
F(x) = -\frac{8}{3} x^{-1/2}.
$$

Differentiate using the power rule:

$$
F'(x) = -\frac{8}{3} \cdot \left(-\frac{1}{2}\right) x^{-3/2} = \frac{8}{6} x^{-3/2} = \frac{4}{3} x^{-3/2}.
$$

Recall that

$$
x^{-3/2} = \frac{1}{x^{3/2}} = \frac{1}{\sqrt{x^3}},
$$

so

$$
F'(x) = \frac{4}{3} \frac{1}{\sqrt{x^3}},
$$

which is exactly the integrand we started with. Thus, the differentiation confirms the solution.

$$
\boxed{-\frac{8}{3\sqrt{x}} + C}
$$
The integral evaluates to $$ -\frac{8}{3\sqrt{x}} + C $$.

] Evaluate the integral. Check your answer by differentiating. (Remember the constant of integration.)