Questions: Can you find the antiderivative for the given problems. Try to do one of them, 1. ∫(5x^4 - 3/x^4 - 6/x^(1/3)) dx 2. ∫x dx / √(2x^2 + 1) 3. ∫(x+2)^4 * x dx

Transcript text: Can you find the antiderivative for the given problems. Try to do one of them, 1. $\int\left(5 x^{4}-\frac{3}{x^{4}}-\frac{6}{\sqrt[3]{x}}\right) \cdot d x$ 2. $\int \frac{x d x}{\sqrt{2 x^{2}+1}}$ 3. $\int(x+2)^{4} \cdot x \cdot d x$

Solution

Solution Steps

To find the antiderivative of a function, we apply the rules of integration. For each term in the integrand, we integrate separately and then combine the results. We use power rule for integration, which states that the integral of $x^n$ is $\frac{x^{n+1}}{n+1}$, and adjust for any constants. For terms involving roots or fractions, we rewrite them in terms of exponents to apply the power rule.

Step 1: Define the Integrand

We start with the integrand for the first problem, which is given by: \[ \int \left(5 x^{4} - \frac{3}{x^{4}} - \frac{6}{\sqrt[3]{x}}\right) \, dx \]

Step 2: Rewrite the Terms

We rewrite the integrand in a more manageable form: \[ 5 x^{4} - 3 x^{-4} - 6 x^{-\frac{1}{3}} \]

Step 3: Apply the Power Rule

We apply the power rule for integration to each term:

The integral of $5 x^{4}$ is $\frac{5}{5} x^{5} = x^{5}$.
The integral of $-3 x^{-4}$ is $-3 \cdot \frac{x^{-3}}{-3} = x^{-3}$.
The integral of $-6 x^{-\frac{1}{3}}$ is $-6 \cdot \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = -9 x^{\frac{2}{3}}$.

Step 4: Combine the Results

Combining all the results, we have: \[ \int \left(5 x^{4} - \frac{3}{x^{4}} - \frac{6}{\sqrt[3]{x}}\right) \, dx = x^{5} + x^{-3} - 9 x^{\frac{2}{3}} + C \] where $C$ is the constant of integration.