Questions: The statement below is of the form [ int f(x) d x=F(x)+C . ] Verify whether the following statement is correct. [ intleft(6 x^2-7 e^xright) d x=3 x^3-7 e^x+C ]

Solution Steps

To verify the given indefinite integral, we need to find the antiderivative of the function \(6x^2 - 7e^x\) and compare it with the provided solution \(3x^3 - 7e^x + C\).

1. Find the antiderivative of \(6x^2\).
2. Find the antiderivative of \(-7e^x\).
3. Combine the results and add the constant of integration \(C\).
4. Compare the result with the given solution.

The antiderivative of \(6x^2\) is calculated as follows: \[ \int 6x^2 \, dx = 2x^3 + C_1 \]

The antiderivative of \(-7e^x\) is: \[ \int -7e^x \, dx = -7e^x + C_2 \]

Combining the results from Steps 1 and 2, we have: \[ \int (6x^2 - 7e^x) \, dx = 2x^3 - 7e^x + C \]

The provided statement was: \[ \int (6x^2 - 7e^x) \, dx = 3x^3 - 7e^x + C \] Comparing the two results, we see that: \[ 2x^3 - 7e^x + C \neq 3x^3 - 7e^x + C \] Thus, the statement is incorrect.

Final Answer