For the following function $$ f $$, find the antiderivative $$ F $$ that satisfies the given condition. $$ f(\mathrm{u})=6 e^{\mathrm{u}}+8 ; F(0)=-8 $$

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To find an antiderivative $$ F(u) $$ of the function $$ f(u) = 6e^{u} + 8 $$ that satisfies the condition $$ F(0) = -8 $$, follow these steps:

1. **Find the General Antiderivative:**

Integrate $$ f(u) $$ with respect to $$ u $$:
   $$
   F(u) = \int f(u) \, du = \int (6e^{u} + 8) \, du
   $$
   $$
   F(u) = 6e^{u} + 8u + C
   $$
   where $$ C $$ is the constant of integration.

2. **Apply the Initial Condition $$ F(0) = -8 $$:**

Substitute $$ u = 0 $$ into the general antiderivative:
   $$
   F(0) = 6e^{0} + 8(0) + C = 6 + 0 + C = 6 + C
   $$
   Set this equal to $$ -8 $$:
   $$
   6 + C = -8
   $$
   $$
   C = -14
   $$

3. **Write the Specific Antiderivative:**

Substitute $$ C = -14 $$ back into the general antiderivative:
   $$
   F(u) = 6e^{u} + 8u - 14
   $$

**Final Answer:**
$$
F(u) = 6\,e^{u} + 8\,u\; -\;14
$$
The antiderivative $$ F(u) $$ is $$ 6e^{u} + 8u - 14 $$.

For the following function \( f \), find the antiderivative \( F \) that satisfies the given condition. \[ f(\mathrm{u})=6 e^{\mathrm{u}}+8 ; F(0)=-8 \]

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