find anitderivative of 30sqroot(x+4)
What is the derivative of (x+4)^(3/2) ?
To find the antiderivative of the function 30√(x+4), we can use the Power Rule for integration combined with the chain rule. The Power Rule states that if we have a function in the form f(x) = x^n, where n is any real number except -1, the antiderivative can be computed by increasing the exponent by 1 and dividing by the new exponent.
In this case, we need to consider the function (x+4)^(1/2). We can rewrite this as (x+4)^(3/2) / (x+4). Now, let's find the derivative of (x+4)^(3/2) using the chain rule.
The chain rule states that if we have a composite function f(g(x)), where g(x) is the inner function and f(u) is the outer function, the derivative can be calculated as f'(g(x)) * g'(x).
Let's apply the chain rule to our function (x+4)^(3/2):
f(g(x)) = (x+4)^(3/2)
u = x+4
Using the power rule, the derivative of f(u) = u^(3/2) is:
f'(u) = (3/2) * u^(3/2 - 1) = (3/2) * u^(1/2) = (3/2) * sqrt(u).
Now, taking into account that u = x+4, we can write the derivative of (x+4)^(3/2) as:
(3/2) * sqrt(x+4).
Therefore, the derivative of (x+4)^(3/2) is (3/2) * sqrt(x+4).
To find the antiderivative of 30√(x+4), we need to reverse the process. Considering the antiderivative as F(x), we have:
F(x) = ∫ 30√(x+4) dx
Since the derivative of (x+4)^(3/2) is (3/2) * sqrt(x+4), we can rewrite the integral as:
F(x) = 30 * ∫ (x+4)^(3/2) dx
Now we can use the Power Rule for integration, which states that if we have a function in the form f(x) = x^n, where n is any real number except -1, the integral can be computed by increasing the exponent by 1 and dividing by the new exponent.
In this case, we increase the exponent 3/2 by 1 to get 5/2 and divide by the new exponent:
F(x) = 30 * (2/5) * (x+4)^(5/2) + C
Where C represents the constant of integration.
Therefore, the antiderivative of 30√(x+4) is 30 * (2/5) * (x+4)^(5/2) + C.