Can someone help me with this question. I'm not sure how to go about it.
Find the antiderivative of
f(x)=2x^3+2x+4
Thanks.
for simple polynomial functions such as this, just integrate each term
so the integral of 2x^3+2x+4
is (2/3)x^3 + x^2 + 4x + C, where C is a constant.
always check your answer by differentiating it, in our case we get back the original expression.
To find the antiderivative of a function, you will need to apply the power rule for integration. The power rule states that for any term in the form of ax^n, where "a" is a constant and "n" is a real number, the antiderivative of that term is (a/(n+1))x^(n+1), plus a constant of integration.
Applying the power rule to each term in the given function f(x) = 2x^3 + 2x + 4 separately, we get:
The antiderivative of 2x^3 is (2/(3+1))x^(3+1) = (1/2)x^4.
The antiderivative of 2x is (2/(1+1))x^(1+1) = x^2.
The antiderivative of 4 is simply 4x.
Now, combining these antiderivatives, we get:
F(x) = (1/2)x^4 + x^2 + 4x + C,
where C is the constant of integration.
Therefore, the antiderivative of f(x) = 2x^3 + 2x + 4 is given by F(x) = (1/2)x^4 + x^2 + 4x + C.