Question

The value of integral \( \int (7x^6 + 1) dx \) is

• A. \( x^7 + k \)
• B. \( 7x^7 + x + k \)
• C. \( x^7 + x + k \)
• D. \( x^6 + x + k \)

Hint:

Don't forget the constant of integration (+k or +C) for indefinite integrals. While it's present in all options here, it's a common point of error in other problems. Remember to integrate each term of a polynomial separately.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question asks for the indefinite integral of a polynomial function. We need to apply the basic rules of integration, specifically the power rule and the rule for integrating a constant.

Step 2: Key Formula or Approach:
The key integration rules needed are:

• Power Rule for Integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) (for \( n \neq -1 \))

• Constant Multiple Rule: \( \int c \cdot f(x) dx = c \cdot \int f(x) dx \)

• Sum Rule: \( \int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx \)

• Integral of a Constant: \( \int c \ dx = cx + C \)
The given integral is \( \int (7x^6 + 1) dx \). We will integrate term by term.

Step 3: Detailed Explanation:
Using the sum rule, we can split the integral into two parts:
\[ \int (7x^6 + 1) dx = \int 7x^6 dx + \int 1 dx \] Now, we integrate each part separately.
For the first term, \( \int 7x^6 dx \):
Using the constant multiple rule, we take the 7 out:
\[ 7 \int x^6 dx \] Now, apply the power rule with n = 6:
\[ 7 \left( \frac{x^{6+1}}{6+1} \right) = 7 \left( \frac{x^7}{7} \right) = x^7 \] For the second term, \( \int 1 dx \):
This is the integral of a constant (c=1):
\[ \int 1 dx = 1 \cdot x = x \] Combine the results:
Now we combine the results of the two integrations and add the constant of integration, denoted here by 'k'.
\[ \int (7x^6 + 1) dx = x^7 + x + k \]

Step 4: Final Answer:
The value of the integral is \( x^7 + x + k \). This matches option (C).