Help me with my Algebra example. Here is a photo

Liwen Guo

Answered question

2022-06-30

RizerMix

Expert2023-05-28Added 656 answers

To find the function f given the derivative

f^{'} (t) = \frac{x + 1}{\sqrt{x}}

and the initial condition f(1) = 5, we can integrate the derivative with respect to x to obtain f(x).
Integrate f'(t) with respect to x:

\int \frac{x + 1}{\sqrt{x}} d x

Simplify the integrand:

\int (x + 1) x^{- 1 / 2} d x

Distribute the integral and split it into two separate integrals:

\int x \cdot x^{- 1 / 2} d x + \int 1 \cdot x^{- 1 / 2} d x

Simplify each integral separately:

\int x \cdot x^{- 1 / 2} d x = \int x^{1 / 2} d x = \frac{2}{3} x^{3 / 2} + C_{1}

\int 1 \cdot x^{- 1 / 2} d x = \int x^{- 1 / 2} d x = 2 x^{1 / 2} + C_{2}

Combine the results and include the constants of integration:

f (x) = \frac{2}{3} x^{3 / 2} + C_{1} + 2 x^{1 / 2} + C_{2}

Apply the initial condition f(1) = 5 to find the values of the constants C1 and C2:

f (1) = \frac{2}{3} (1^{3 / 2}) + C_{1} + 2 (1^{1 / 2}) + C_{2} = \frac{2}{3} + C_{1} + 2 + C_{2} = 5

Solve for the constants C1 and C2:

\frac{2}{3} + C_{1} + 2 + C_{2} = 5

Combine like terms:

C_{1} + C_{2} + \frac{8}{3} = 5

Subtract

\frac{8}{3}

from both sides:

C_{1} + C_{2} = 5 - \frac{8}{3}

C_{1} + C_{2} = \frac{7}{3}

Substitute the values of C1 and C2 back into the equation for f(x):

f (x) = \frac{2}{3} x^{3 / 2} + \frac{7}{3}

Therefore, the function f(x) is

f (x) = \frac{2}{3} x^{3 / 2} + \frac{7}{3}

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