Question

How do you find the antiderivative of $f\left( x \right) = \left( {2x + 3} \right)\left( {3x - 7} \right)$ ?

Answer 1

In calculus, anti derivative of a function is same as indefinite integral of the given function. The given question requires us to integrate a function of x with respect to x. So, we will first find the product of the two terms given in the function and then integrate using the power rule of integration.

Complete step by step answer:
The given question requires us to integrate a product function $f\left( x \right) = \left( {2x + 3} \right)\left( {3x - 7} \right)$ in variable $x$ whose both brackets are linear polynomials in x and consists of two terms. So, we first simplify the function by opening brackets and then integrate the function directly using the power rule of integration. So, we have, $f\left( x \right) = \left( {2x + 3} \right)\left( {3x - 7} \right)$

So, we multiply both the terms of the first bracket with the entire second bracket. So, we get,
$\Rightarrow f\left( x \right) = 2x\left( {3x - 7} \right) + 3\left( {3x - 7} \right)$
Expanding the expression by opening the brackets, we get,
$\Rightarrow f\left( x \right) = 6{x^2} - 14x + 9x - 21$
Adding the like terms, we get,
$\Rightarrow f\left( x \right) = 6{x^2} - 5x - 21$
Now, we integrate this function. So, we have to evaluate $\int {\left( {6{x^2} - 5x - 21} \right)} dx$.

Now, we know the power rule of integration. According to the power rule of integration, the integral of ${x^n}$ with respect to x is $\dfrac{{{x^{\left( {n + 1} \right)}}}}{{n + 1}}$.
So, we get, $\int {\left( {6{x^2} - 5x - 21} \right)} dx = \left( {6\dfrac{{{x^{2 + 1}}}}{{2 + 1}} - 5\dfrac{{{x^{1 + 1}}}}{{1 + 1}} - 21\dfrac{{{x^{0 + 1}}}}{{0 + 1}}} \right) + c$
$\Rightarrow \int {\left( {6{x^2} - 5x - 21} \right)} dx = \left( {6\dfrac{{{x^3}}}{3} - 5\dfrac{{{x^2}}}{2} - 21x} \right) + c$
Cancelling common factors in numerator and denominator, we get,
$\Rightarrow \int {\left( {6{x^2} - 5x - 21} \right)} dx = 2{x^3} - \dfrac{5}{2}{x^2} - 21x + c$, where c is the constant of integration.

So, $2{x^3} - \dfrac{5}{2}{x^2} - 21x + c$ is the antiderivative of the given function $f\left( x \right) = \left( {2x + 3} \right)\left( {3x - 7} \right)$.

Note: The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant. The product of two binomial linear expressions can be computed using distributive property $a\left( {b + c} \right) = ab + ac$.