For the following function $$ f $$, find the antiderivative $$ F $$ that satisfies the given condition $$ f(\mathrm{u})=5 e^{\mathrm{u}}+12 ; \mathrm{F}(0)=-7 $$ The antiderivative that satisfies the given condition is $$ \mathrm{F}(\mathrm{u})=\square $$.

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To find the antiderivative $$ F(u) $$ of the function $$ f(u) = 5 e^{u} + 12 $$ that satisfies the condition $$ F(0) = -7 $$, follow these steps:

1. **Find the General Antiderivative:**

Integrate $$ f(u) $$ with respect to $$ u $$:
   $$
   F(u) = \int f(u) \, du = \int (5 e^{u} + 12) \, du
   $$
   $$
   F(u) = 5 e^{u} + 12 u + C
   $$
   where $$ C $$ is the constant of integration.

2. **Apply the Initial Condition:**

Use the condition $$ F(0) = -7 $$ to solve for $$ C $$:
   $$
   F(0) = 5 e^{0} + 12 \cdot 0 + C = 5(1) + 0 + C = 5 + C
   $$
   $$
   5 + C = -7 \implies C = -12
   $$

3. **Write the Specific Antiderivative:**

Substitute $$ C = -12 $$ back into the general antiderivative:
   $$
   F(u) = 5 e^{u} + 12 u - 12
   $$

**Final Answer:**
$$
F(u) = 5\, e^{\,u} + 12\,u\ -\ 12
$$
The antiderivative is $$ F(u) = 5e^{u} + 12u - 12 $$.

For the following function \( f \), find the antiderivative \( F \) that satisfies the given condition \( f(\mathrm{u})=5 e^{\mathrm{u}}+12 ; \mathrm{F}(0)=-7 \) The antiderivative that satisfies the given condition is \( \mathrm{F}(\mathrm{u})=\square \).

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