# Evaluating a definite integral

#### Pranav

##### Well-known member
Problem:
If f is continuous and differentiable function in $x \in (0,1)$ suuch that $\sum_{r=0}^{1}\left(f(x+r)-\left|e^x-r-1\right|\right)$=0, then $\int_0^{11} f(x)\,dx$ is

A)65+4ln2-7e
B)63+4ln2-9e
C)69-9e
D)29-23e

Ans: A

Attempt:
I could only write the following:
$$f(x)+f(x+1)+\cdots+f(x+11)=|e^x-1|+|e^x-2|+\cdots+|e^x-11|$$
Since I had no idea how to proceed further, I assumed $f(x)=|e^x-11|$ but evaluating the definite integral with this f(x) doesn't give the right answer.

Any help is appreciated. Thanks!

#### ZaidAlyafey

##### Well-known member
MHB Math Helper
Hint: you can write the integral as

$$\displaystyle \int^1_0+\int^2_1 + \cdots +\int^{11}_{10} f(x) \, dx$$

Then $$\displaystyle \int^2_1 f(x) \, dx = \int^1_0 f(x+1) \, dx$$

#### Pranav

##### Well-known member
Hint: you can write the integral as

$$\displaystyle \int^1_0+\int^2_1 + \cdots +\int^{11}_{10} f(x) \, dx$$

Then $$\displaystyle \int^2_1 f(x) \, dx = \int^1_0 f(x+1) \, dx$$
Ah, how could I miss that. Thanks a lot ZaidAlyafey! 