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  • Need solution for RD sharma maths class 12 chapter 19 Definite Integrals exercise 19.5 question 11

Need solution for RD sharma maths class 12 chapter 19 Definite Integrals exercise 19.5 question 11

Answers (1)

\frac{4}{3}

Hint:

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

Given:

\int_{1}^{2}\left ( x^{2}-1 \right )dx

Solution:

We have,

\int_{a}^{b}\left ( fx \right )dx=\lim_{h\rightarrow 0}h\left [ f(a)+f(a+h)+f(a+2h)+.....f(a+(n-1)h) \right ]

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

Here,

a=1,b=2,f(x)=x^{2}-1\\ h=\frac{2-1}{n}=\frac{1}{n}

Thus, we have

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

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

\begin{aligned} &=\lim _{n \rightarrow \infty} \frac{1}{n}\left[\frac{2}{n}\left(\frac{n(n-1)}{2}\right)+\frac{1}{n^{2}}\left(\frac{n(n-1)(2 n-1)}{6}\right)\right] \quad\left[\begin{array}{l} \mathrm{Q} h=\frac{1}{n} \text { if } \\ h \rightarrow 0, n \rightarrow \infty \end{array}\right] \\ \end{aligned}

\begin{aligned} &=\lim _{n \rightarrow \infty} \frac{1}{n^{2}} \cdot n^{2}\left(1-\frac{1}{n}\right)+\frac{1}{6 n^{3}} \cdot n^{3}\left(1-\frac{1}{n}\right)\left(2-\frac{1}{n}\right) \\ \end{aligned}

\begin{aligned} &=1+\frac{2}{6} \\ &=\frac{6+2}{6}=\frac{8}{6} \\ &=\frac{4}{3} \end{aligned}

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