Answer :
The integral of ∫(7x⁶ - 4x⁹ + 3)dx is found by applying the power rule for integration to each term, resulting in x⁷ - (2/5)x¹⁰ + 3x + C, where C is the constant of integration.
To evaluate the integral ∫(7x⁶ - 4x⁹ + 3)dx, we use rules of antiderivatives. Each term of the polynomial is integrated separately according to the power rule for integration, which states that the integral of x raised to a power n (where n is not -1) is x to the power of (n+1) divided by (n+1), plus a constant. We apply this rule term-by-term:
- For 7x⁶, the antiderivative is 7 times the integral of x⁶, which is (7/7)x⁷ or x⁷.
- For -4x⁹, the antiderivative is -4 times the integral of x⁹, which is (-4/10)x¹⁰ or -2/5x¹⁰.
- For the constant 3, the antiderivative is 3x, since the integral of a constant is the constant multiplied by the variable of integration.
Combining these results, the integral of the given function is x⁷ - (2/5)x¹⁰ + 3x + C, where C represents the constant of integration.
