Question

How do you find the antiderivative of \[f(x) = 4{x^2} - 6x + 7\] ?

Answer 1

Hint: We use the general formula of integration and solve the antiderivative of the given function. Separate the integral to each value and take out the constant terms from the integration. Use the formula of integration in each term.
* Anti-derivative of a function is the opposite of the derivative of a function i.e. it is that value which can be obtained when taking reverse of the derivative function. It is that function whose derivative we take and then we get the function.
* If \[y = {x^n}\]then integration of y with respect to x will be given by the formula: \[\int {ydx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C\] , where C is constant of integration

Complete step by step solution:
We have to find the antiderivative of \[f(x) = 4{x^2} - 6x + 7\]
We have to solve for the value of \[\int {f(x)dx} \]
i.e. we have to calculate \[\int {4{x^2} - 6x + 7} dx\] … (1)
We can separate the integration of complete function f(x) into integration of each term along with the sign in between the integrals.
\[ \Rightarrow \int {4{x^2} - 6x + 7} dx = \int {4{x^2}dx} - \int {6xdx} + \int {7dx} \]
Now we bring out respective constant terms from all integrals on the right hand side of the equation.
\[ \Rightarrow \int {4{x^2} - 6x + 7} dx = 4\int {{x^2}dx} - 6\int {xdx} + 7\int {dx} \]
Now we know the formula of integration: \[\int {ydx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C\] where C is constant of integration
\[ \Rightarrow \int {4{x^2} - 6x + 7} dx = 4\left( {\dfrac{{{x^{2 + 1}}}}{{2 + 1}}} \right) - 6\left( {\dfrac{{{x^{1 + 1}}}}{{1 + 1}}} \right) + 7\left( {\dfrac{{{x^{0 + 1}}}}{{0 + 1}}} \right) + C\]
Add the values in the power of ‘x’ and the values in the denominators for each term separately.
\[ \Rightarrow \int {4{x^2} - 6x + 7} dx = 4\left( {\dfrac{{{x^3}}}{3}} \right) - 6\left( {\dfrac{{{x^2}}}{2}} \right) + 7\left( {\dfrac{{{x^1}}}{1}} \right) + C\]
Cancel possible factors from the numerator and denominator of each fraction separately.
\[ \Rightarrow \int {4{x^2} - 6x + 7} dx = \dfrac{{4{x^3}}}{3} - 3{x^2} + 7x + C\]
\[\therefore \] The anti-derivative of \[f(x) = 4{x^2} - 6x + 7\] is \[\dfrac{{4{x^3}}}{3} - 3{x^2} + 7x + C\]

Note: Many students make the mistake of calculating the derivative first and then integrate the obtained derivative which is wrong, keep in mind we have to calculate the integration of the function, not the derivative. Anti-derivative of a function means that value which on differentiation or derivative gives the function. Also, many students tend to ignore the addition of constant term in the end, keep in mind we always add constant term when integrating without limits.