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  • Provide solution for RD sharma maths class 12 chapter 20 Areas Of Bounded Region exercise Fill in the blanks question 7

Provide solution for RD sharma maths class 12 chapter 20 Areas Of Bounded Region exercise Fill in the blanks question 7

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Hint:

Use indefinite integral formula then put limits to solve this integral

Given:

y=2^{kx},x=0,x=2\\

Area= 3 \log_{2}e

Explanation:Area \begin{aligned} =\int_{0}^{2} 2^{k x} d x \quad\left[\because \int a^{x} d x=\frac{a^{x}}{\log _{e} a}\right] \\ \end{aligned}

\begin{aligned} &3 \log _{2} e=\left[\frac{1}{k} \frac{2^{k x}}{\log _{e} 2}\right]_{0}^{2} \\ \end{aligned}

\begin{aligned} &3 \log _{2} e=\left[\frac{\log _{2} e}{k} 2^{k x}\right]_{0}^{2} \quad\left[\because \log _{a} e=\frac{1}{\log _{e} a}\right] \\ \end{aligned}

\begin{aligned} &3 k \log _{2} e=\log _{2} e\left[2^{k x}\right]_{0}^{2} \\ &3 k=\left(2^{k}\right)^{2}-\left(2^{k}\right)^{0} \\ &3 k=2^{2 k}-1 \\ &2^{2 k}-3 k=1 \end{aligned}

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