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  • please solve RD sharma class 12 chapter 19 Definite Integrals exercise 19.5 question 2 maths textbook solution

please solve RD sharma class 12 chapter 19 Definite Integrals exercise 19.5 question 2 maths textbook solution

Answers (1)

8

Hint:

To solve the given statement, we have to use the formula of addition limits.

Given:

\int_{0}^{2}(x+3) d x

Solution:

We have,

\int_{a}^{b}(f x) d x=\lim _{h \rightarrow 0} h[f(a)+f(a+h)+f(a+2 h)+\ldots f(a+(n-1) h)]

Where, h=\frac{b-a}{n}

Here,

\begin{aligned} &a=0, b=2, f(x)=(x+3) \\ &h \rightarrow \frac{2}{n} \end{aligned}

Thus, we have

\begin{aligned} I &=\int_{0}^{2}(x+3) d x \\ I &=\lim _{h \rightarrow 0} h[f(0)+f(h)+f(2 h)+\ldots f(n-1) h] \\ \end{aligned}

\begin{aligned} &=\lim _{h \rightarrow 0} h[3+(h+3)+(2 h+3)+\ldots((n-1) h+3)] \\ \end{aligned}

\begin{aligned} &=\lim _{h \rightarrow 0} h[3 n+h(1+2+3+\ldots(n-1))] \\ \end{aligned}

\begin{aligned} &=\lim _{h \rightarrow 0} h\left[3 n+h\left(\frac{n(n-1)}{2}\right)\right] \\ \end{aligned}

\begin{aligned} &=\lim _{n \rightarrow \infty} \frac{2}{n}\left[3 n+\frac{2}{n} \frac{n(n-1)}{2}\right] & &&&&&\quad\left[\begin{array}{l} \left.\mathrm{Q} h \rightarrow 0 \& h=\frac{2}{n}\right] \\ n \rightarrow \infty \end{array}\right] \\ \end{aligned}

\begin{aligned} &=\lim _{n \rightarrow \infty}\left[6+\frac{2}{n} \cdot n^{2}\left(1-\frac{1}{n}\right)\right] \\ &=6+2=8 \end{aligned}

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