Evaluate the indefinite integral: ${\int }_{}^{}(x^3-8x^2-7x+10)dx$

• A. $\frac{1}{3}{x}^3-4x^2-7x+10+C$
• B. $\frac{1}{4}{x}^4-\left(\frac{8}{3}\right)x^3-\left(\frac{7}{2}\right)x^2+10x+C$
• C. $x^4-8x^3-7x^2+10x+C$
• D. $\frac{1}{4}{x}^4-8x^2-7x+10x+C$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the formula for the indefinite integral of a polynomial term. For a term of the form x^n, the integral is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
2. Step 2: Break the given polynomial into individual terms: x^3, -8x^2, -7x, and 10. Each term will be integrated separately.
3. Step 3: Apply the integration formula to each term: For x^3, the integral is (1/4)x^4. For -8x^2, the integral is (-8/3)x^3. For -7x, the integral is (-7/2)x^2. For the constant 10, the integral is 10x.
4. Step 4: Combine all the integrated terms into a single expression: (1/4)x^4 - (8/3)x^3 - (7/2)x^2 + 10x + C.
5. Step 5: Verify the result by differentiating the combined expression to ensure it matches the original polynomial. Differentiation of (1/4)x^4 - (8/3)x^3 - (7/2)x^2 + 10x + C should yield x^3 - 8x^2 - 7x + 10.