To solve the problem, we need to evaluate the expression:
\[
\int_1^5 f(x) \, dx + \int_2^{10} g(y) \, dy
\]
where \( g(x) \) is the inverse of \( f(x) \).
### Step 1: Understand the relationship between \( f(x) \) and \( g(y) \)
Since \( g(y) \) is the inverse of \( f(x) \), we have:
- \( y = f(x) \) implies \( x = g(y) \).
### Step 2: Change the variable in the second integral
We know that if \( y = f(x) \), then \( dy = f'(x) \, dx \). The limits of integration change as follows:
- When \( y = 2 \), \( x = g(2) \) which corresponds to \( f(1) = 2 \) so \( x = 1 \).
- When \( y = 10 \), \( x = g(10) \) which corresponds to \( f(5) = 10 \) so \( x = 5 \).
Thus, we can rewrite the second integral:
\[
\int_2^{10} g(y) \, dy = \int_1^5 g(f(x)) f'(x) \, dx
\]
Since \( g(f(x)) = x \), we have:
\[
\int_2^{10} g(y) \, dy = \int_1^5 x f'(x) \, dx
\]
### Step 3: Combine the integrals
Now, we can combine both integrals:
\[
\int_1^5 f(x) \, dx + \int_1^5 x f'(x) \, dx
\]
### Step 4: Use integration by parts on the second integral
Let \( u = x \) and \( dv = f'(x) \, dx \). Then \( du = dx \) and \( v = f(x) \). Using integration by parts:
\[
\int x f'(x) \, dx = x f(x) - \int f(x) \, dx
\]
### Step 5: Evaluate the combined integral
Substituting back, we have:
\[
\int_1^5 f(x) \, dx + \left[ x f(x) - \int_1^5 f(x) \, dx \right]_1^5
\]
This simplifies to:
\[
\int_1^5 f(x) \, dx + \left[ x f(x) \right]_1^5
\]
### Step 6: Calculate the limits
Now we evaluate \( \left[ x f(x) \right]_1^5 \):
\[
= 5 f(5) - 1 f(1) = 5 \cdot 10 - 1 \cdot 2 = 50 - 2 = 48
\]
### Step 7: Final result
Thus, the final result is:
\[
\int_1^5 f(x) \, dx + 48
\]
Since \( \int_1^5 f(x) \, dx \) cancels out with the negative part from integration by parts, we conclude that:
\[
\int_1^5 f(x) \, dx + \int_2^{10} g(y) \, dy = 48
\]
### Final Answer
The value of \( \int_1^5 f(x) \, dx + \int_2^{10} g(y) \, dy \) equals **48**.
