To solve the integral of $\int (x^2 - 7x + 5) \, dx$, follow these steps:

Step 1: Split the Integral

We can separate the terms of the polynomial: $\int (x^2 - 7x + 5) \, dx = \int x^2 \, dx - \int 7x \, dx + \int 5 \, dx$

Step 2: Integrate Each Term

1. For $\int x^2 \, dx$: $\int x^2 \, dx = \frac{x^3}{3}$

2. For $\int 7x \, dx$: $\int 7x \, dx = \frac{7x^2}{2}$

3. For $\int 5 \, dx$: $\int 5 \, dx = 5x$

Step 3: Combine the Results

Adding these together, the indefinite integral is: $\int (x^2 - 7x + 5) \, dx = \frac{x^3}{3} - \frac{7x^2}{2} + 5x + C$

Here, $C$ is the constant of integration.

Do you want a detailed explanation of any part? Or perhaps you'd like to verify this with specific limits of integration?

Related Questions:

1. How do you integrate higher-degree polynomials like $x^3 - 4x^2 + 6x$?
2. What changes when we compute definite integrals instead of indefinite ones?
3. How is the constant of integration interpreted in practical problems?
4. How would the integral change if the coefficients were functions instead of constants?
5. Can integration be applied to trigonometric or exponential terms in a similar way?

Tip:

When integrating polynomials, remember to increment the power by 1 and divide by the new power for each term!