If $$ \int_{2}^{2-x} f(t) d t=\sqrt{1+\sin (x)}+c $$, then $$ c= $$

Question

UpStudy AI · Accepted Answer

To find the constant $$ c $$ in the equation:

$$
\int_{2}^{2-x} f(t) \, dt = \sqrt{1 + \sin(x)} + c
$$

we can use a strategic value for $$ x $$ that simplifies the equation. A natural choice is $$ x = 0 $$:

1. **Evaluate the integral when $$ x = 0 $$:**
   
   $$
   \int_{2}^{2-0} f(t) \, dt = \int_{2}^{2} f(t) \, dt = 0
   $$

2. **Evaluate the right-hand side when $$ x = 0 $$:**
   
   $$
   \sqrt{1 + \sin(0)} + c = \sqrt{1 + 0} + c = 1 + c
   $$

3. **Set the two expressions equal and solve for $$ c $$:**
   
   $$
   0 = 1 + c \quad \Rightarrow \quad c = -1
   $$

**Answer:** $$ -1 $$
$$ c = -1 $$

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