First slide

Evaluation of definite integrals

Question

The value of integral $\sum_{k = 1}^{n} \int_{0}^{1} f (k - 1 + x) dx$ is

Moderate

A

$\int_{0}^{1} (x) dx$
B

$\int_{0}^{2} f (x) dx$
C

$\int_{0}^{n} f (x) dx$
D

$n \int_{0}^{1} f (x) dx$

Solution

$\begin{array}{ll} Let I = \int_{0}^{1} f (k - 1 + x) dx \\ \Rightarrow I = \int_{k - 1}^{k} f (t) dt, where t = k - 1 + x \\ \Rightarrow I = \int_{k - 1}^{k} f (x) dx \\ ∴ \sum_{k = 1}^{n} \int_{k - 1}^{k} f (x) dx \\ = \int_{0}^{1} f (x) dx + \int_{1}^{2} f (x) dx + \dots + \int_{n - 1}^{n} f (x) dx = \int_{0}^{n} f (x) dx \end{array}$

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