To solve the integral:

$\int (3x^5 - 7x + 3) \, dx$

we'll integrate each term in the expression separately.

Step-by-Step Solution

1. Integral of $3x^5$: $\int 3x^5 \, dx = 3 \cdot \frac{x^{5+1}}{5+1} = \frac{3x^6}{6} = \frac{x^6}{2}$

2. Integral of $-7x$: $\int -7x \, dx = -7 \cdot \frac{x^{1+1}}{1+1} = -7 \cdot \frac{x^2}{2} = -\frac{7x^2}{2}$

3. Integral of $3$: $\int 3 \, dx = 3x$

Combining the Results

Now, we combine these results:

$\int (3x^5 - 7x + 3) \, dx = \frac{x^6}{2} - \frac{7x^2}{2} + 3x + C$

where $C$ is the constant of integration.

Final Answer

$\int (3x^5 - 7x + 3) \, dx = \frac{x^6}{2} - \frac{7x^2}{2} + 3x + C$

Would you like further details on any of the steps?

Related Questions

1. How do you integrate a polynomial with multiple terms?
2. What is the role of the constant of integration, $C$, in indefinite integrals?
3. How does the power rule work for integration?
4. What are the differences between definite and indefinite integrals?
5. How would the answer change if this were a definite integral over a specific interval?

Tip

When integrating a polynomial, remember that each term can be integrated individually, and the constant $C$ must be included in indefinite integrals.