If f is the function defined by f(x)=(x^2+4x)^(1/3) and g is an antiderivative of f such that g(5)=7, then g(1) Is congruent to

To find the value of g(1), we need to use the Fundamental Theorem of Calculus. According to the theorem, if g is an antiderivative of f, then the definite integral of f from a to b is equal to g(b) - g(a).

Since g(5) = 7, we can say that the definite integral of f from 5 to 1 is equal to g(1) - g(5):
∫[5 to 1] f(x) dx = g(1) - g(5)

Now, let's find the definite integral of f(x).
f(x) = (x^2 + 4x)^(1/3)

To integrate this function, we can use the substitution method:
Let u = x^2 + 4x
Therefore, du = (2x + 4) dx or (dx/2) = du/(2x + 4)

When x = 5, u = 5^2 + 4(5) = 45
When x = 1, u = 1^2 + 4(1) = 9

Now, let's substitute the bounds:
∫[5 to 1] f(x) dx = ∫[45 to 9] u^(1/3) * (dx/2)

Let's simplify this integral:
(1/2) ∫[45 to 9] u^(1/3) du

To integrate u^(1/3), we add 1 to the exponent and divide by the new exponent:
(1/2) * (3/4) * u^(4/3) | [45 to 9]

Now, substitute the values of u:
(1/2) * (3/4) * (9^(4/3) - 45^(4/3))

Evaluating this expression will give us the definite integral of f(x) from 5 to 1. Subtracting g(5) = 7 from this value will give us g(1):
g(1) = [(1/2) * (3/4) * (9^(4/3) - 45^(4/3))] - 7

To find the value of g(1), we need to evaluate the indefinite integral of f(x) and use the given information to determine a particular value.

Let's start by finding the antiderivative of f(x):

Integrate f(x) with respect to x:
∫ [(x^2 + 4x)^(1/3)] dx

Using the substitution method, let's substitute u = x^2 + 4x:
u = x^2 + 4x
du/dx = 2x + 4
dx = du/(2x + 4)

The integral becomes:
∫ [(u)^(1/3)] * (du/(2x + 4))

Now, we need to change the limits of integration based on the substitution:
When x = 5, u = 5^2 + 4(5) = 25 + 20 = 45
When x = 1, u = 1^2 + 4(1) = 1 + 4 = 5

Substituting the limits of integration and simplifying, we have:
∫ [(u)^(1/3)] * (du/(2x + 4)) evaluated from u=45 to u=5

Now we can integrate using the power rule for integration:
1/4 * (u^(4/3)) evaluated from u=45 to u=5

Applying the power rule, we have:
1/4 * (5^(4/3) - 45^(4/3))

Calculating the values:
1/4 * (125^(1/3) - 50625^(1/3))
1/4 * (5 - 45)
1/4 * (-40)
-10

Therefore, g(1) is congruent to -10.

1.710, to be honest not sure if its right but I just equalled x to 1 as it wanted and found where it intersected the f(x) graph and just went with that tbh.