Questions: Evaluate the indefinite integrals in Exercises 1-8. 1. ∫(3x^2+2x+x^(-3)) dx

Transcript text: Evaluate the indefinite integrals in Exercises 1-8. 1. $\int\left(3 x^{2}+2 x+x^{-3}\right) d x$

Solution

Solution Steps

To evaluate the indefinite integral, we need to integrate each term of the polynomial separately. The integral of $x^n$ is $\frac{x^{n+1}}{n+1}$ for any real number $n \neq -1$. We will apply this rule to each term in the integrand.

Step 1: Integrate Each Term Separately

To evaluate the indefinite integral $\int (3x^2 + 2x + x^{-3}) \, dx$, we integrate each term separately using the power rule for integration.

Step 2: Apply the Power Rule

The power rule for integration states that $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ for any real number $n \neq -1$.

For the term $3x^2$: \[ \int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = x^3 \]
For the term $2x$: \[ \int 2x \, dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = x^2 \]
For the term $x^{-3}$: \[ \int x^{-3} \, dx = \frac{x^{-3+1}}{-3+1} = \frac{x^{-2}}{-2} = -\frac{1}{2} x^{-2} \]

Step 3: Combine the Results

Combine the integrals of each term to get the final result: \[ \int (3x^2 + 2x + x^{-3}) \, dx = x^3 + x^2 - \frac{1}{2} x^{-2} + C \]