find the general antiderivative of (f(x)=42x^{2}) and check the answer by differentiating. (use symbolic…

Explanation:

Step1: Apply power - rule for antiderivatives

The power - rule for antiderivatives is $\int x^n dx=\frac{x^{n + 1}}{n+1}+C$, where $n\neq - 1$. For $f(x)=42x^2$, we have: $\int42x^2dx=42\int x^2dx$

Step2: Calculate the antiderivative

Using the power - rule with $n = 2$, we get $42\times\frac{x^{2+1}}{2 + 1}+C$. $42\times\frac{x^{3}}{3}+C=14x^{3}+C$

Step3: Check by differentiating

Differentiate $F(x)=14x^{3}+C$ using the power - rule for differentiation $\frac{d}{dx}(x^n)=nx^{n - 1}$. $\frac{d}{dx}(14x^{3}+C)=14\times3x^{2}+0 = 42x^{2}$

Answer:

$14x^{3}+C$