Question

Find the particular antiderivative that satisfies the following conditions:

Fʹ(x)=x⁵+2 √(x) ; F(1)=-7

F(x)=

Answer 1

The particular antiderivative F(x) that satisfies the given conditions is:

F(x) = (1/6)x^6 + (4/3)x^(3/2) - 26/3

Step-by-step explanation:

We integrate the function F'(x) = x^5 + 2√(x).

Integrating term by term, we have:

∫(x^5 + 2√(x)) dx = ∫x^5 dx + ∫2√(x) dx

Integrating each term separately:

= (1/6)x^6 + 2(2/3)(x^3/2) + C

= (1/6)x^6 + (4/3)x^(3/2) + C

Where C is the constant of integration. Now, we need to determine the value of C using the given condition F(1) = -7.

Plugging x = 1 into the expression for F(x), we have:

F(1) = (1/6)(1^6) + (4/3)(1^(3/2)) + C

= 1/6 + 4/3 + C

= 2/6 + 8/6 + C

= 10/6 + C

= 5/3 + C

Since F(1) is given to be -7, we set 5/3 + C = -7 and solve for C:

5/3 + C = -7

C = -7 - 5/3

C = -21/3 - 5/3

C = -26/3

F(x) = (1/6)x^6 + (4/3)x^(3/2) - 26/3

Answer 2

The particular antiderivative that satisfies the given conditions is F(x) = (1/6)x^6 + (4/3)x^(3/2) - 8.

Step-by-step explanation:

To find the particular antiderivative that satisfies the given conditions, we can integrate the given function with respect to x. Let's start by integrating each term separately. The integral of x^5 is (1/6)x^6, and the integral of 2√x is (4/3)x^(3/2). Adding these two integrals together, we get F(x) = (1/6)x^6 + (4/3)x^(3/2) + C, where C is the constant of integration.

Now, we can use the given initial condition to find the value of the constant C. Substituting x=1 and F(x)=-7 into the equation, we get -7 = (1/6)(1)^6 + (4/3)(1)^(3/2) + C. Solving for C, we find C = -8. Therefore, the particular antiderivative that satisfies the given conditions is F(x) = (1/6)x^6 + (4/3)x^(3/2) - 8.