Name: Integration by Reduction Formula
Uploaded: 2019-07-19
Duration: 10 min 19 s
Description: A reduction formula is one that enables us to solve an integral problem by reducing it to a problem of solving an easier integral problem. Watch this video t...

Question

Evaluate the following integral as limit of a sum:

\int_{0}^{5} (x + 1) d x

.

Accepted Answer

Let $I = \int_{0}^{5} (x + 1) d x$

We know that, $\int_{a}^{b} f (x) d x = (b - a) \lim_{n \to \infty} \frac{1}{n} [f (a) + f (a + h) + \dots + f (a + (n - 1) h)]$ , where $h = \frac{b - a}{n}$ .

Here, $a = 0, b = 5$ and $f (x) = x + 1$

$\Rightarrow h = \frac{5 - 0}{n} = \frac{5}{n}$

$∴ \int_{0}^{5} (x + 1) d x = (5 - 0) \lim_{n \to \infty} \frac{1}{n} [f (0) + f (\frac{5}{n}) + \dots + f ((n - 1) \frac{5}{n})]$

$= 5 \lim_{n \to \infty} \frac{1}{n} [1 + (\frac{5}{n} + 1) + \dots \{1 + (\frac{5 (n - 1)}{n})\}]$

$= 5 \lim_{n \to \infty} \frac{1}{n} [(1 + 1 + 1 \dots 1) + [\frac{5}{n} + 2 \cdot \frac{5}{n} + 3 \cdot \frac{5}{n} + \dots (n - 1) \frac{5}{n}]]$

$= 5 \lim_{n \to \infty} \frac{1}{n} [n + \frac{5}{n} {1 + 2 + 3 \dots (n - 1)}]$

$= 5 \lim_{n \to \infty} \frac{1}{n} [n + \frac{5}{n} \cdot \frac{(n - 1) n}{2}] [∵ \sum_{}^{} n = \frac{n (n + 1)}{2}]$

$= 5 \lim_{n \to \infty} [1 + \frac{5}{2} (1 - \frac{1}{n})]$

$= 5 [1 + \frac{5}{2}] [∵ \lim_{n \to \infty} \frac{1}{n} = 0]$

$= 5 [\frac{7}{2}]$

$= \frac{35}{2}$

Hence, $\int_{0}^{5} (x + 1) d x = \frac{35}{2}$ .

Telangana Board Solutions for Exercise 1: Exercise 7(a)

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Exercise 1: Exercise 7(a)

Questions from Telangana Board Solutions for Exercise 1: Exercise 7(a) with Hints & Solutions

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