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  • Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 18

Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 18

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Answer:   \frac{\sqrt{e}-1}{e}

Given:  \int_{1}^{2} \frac{1}{x^{2}} e^{-\frac{1}{x}} d x

Hint: Use Substitution method

Solution:

Let

\begin{aligned} &\frac{-1}{x}=t \\ & \end{aligned}                  (Differentiating w.r.t to x)

\frac{1}{x^{2}} d x=d t

\begin{aligned} &\int_{1}^{2} \frac{1}{x^{2}} e^{-\frac{1}{x}} d x\\ & \end{aligned}

\int_{1}^{2} e^{t} d t=\left(e^{t}\right)_{1}^{2}=\left(e^{\frac{-1}{x}}\right)_{1}^{2}\\

\begin{aligned} &=e^{\frac{-1}{2}}-e^{-1} \\ \end{aligned}

=e^{-\frac{1}{2}}-\frac{1}{e} \\

=\frac{\sqrt{e}-1}{e}

 

 

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