How do you find the antiderivative of #f(x)=(2x+3)(3x-7)#?

Factor out the expression.

The antiderivative is as follows

To find the antiderivative of ( f(x) = (2x + 3)(3x - 7) ), you can use the distributive property to expand the expression and then integrate each term separately. The antiderivative of a constant multiplied by a variable raised to a power can be found using the power rule of integration. After integrating each term, you combine them to find the overall antiderivative.

Here are the steps:

1. Expand the expression: ( f(x) = 6x^2 - 14x + 9x - 21 ).
2. Combine like terms: ( f(x) = 6x^2 - 5x - 21 ).
3. Integrate each term separately:
   • The antiderivative of ( 6x^2 ) is ( \frac{6}{3}x^3 = 2x^3 ).
   • The antiderivative of ( -5x ) is ( -\frac{5}{2}x^2 ).
   • The antiderivative of ( -21 ) is ( -21x ).
4. Combine the antiderivatives: ( 2x^3 - \frac{5}{2}x^2 - 21x + C ), where ( C ) is the constant of integration.

To find the antiderivative of ( f(x) = (2x+3)(3x-7) ), we can use the distributive property to expand the expression, integrate each term separately, and then sum the results. The antiderivative of a constant multiplied by a function is the constant times the antiderivative of the function. Applying these principles, we integrate each term separately:

[ \int (2x+3)(3x-7) , dx = \int (6x^2 - 14x + 9x - 21) , dx ]

[ = \int (6x^2 - 5x - 21) , dx ]

Now, we integrate each term separately:

[ \int 6x^2 , dx = 2x^3 + C_1 ]

[ \int -5x , dx = - \frac{5}{2} x^2 + C_2 ]

[ \int -21 , dx = -21x + C_3 ]

Where ( C_1, C_2, ) and ( C_3 ) are constants of integration.

Finally, we combine these results:

[ \int (2x+3)(3x-7) , dx = 2x^3 - \frac{5}{2} x^2 - 21x + C ]

Where ( C ) represents the constant of integration.