Mathematics

Evaluate: $$\int\limits_1^2 {\dfrac{1}{{{x^2}}}{e^{\tfrac{{ - 1}}{x}}}dx = } $$


ANSWER

$$\dfrac{{\sqrt e - 1}}{e}$$


SOLUTION
$$\displaystyle\int_{1}^{2}{\dfrac{1}{{x}^{2}}{e}^{-\frac{1}{x}}dx}$$

Let $$t=-\dfrac{1}{x}\Rightarrow\,dt=-\left(-\dfrac{1}{{x}^{2}}\right)dx=\dfrac{1}{{x}^{2}}dx$$

When $$x=1\Rightarrow\,t=-1$$

When $$x=2\Rightarrow\,t=-\dfrac{1}{2}$$

$$=\displaystyle\int_{-1}^{-\tfrac{1}{2}}{{e}^{t}dt}$$

$$=\left[{e}^{t}\right]_{-1}^{-\tfrac{1}{2}}$$

$$={e}^{-\tfrac{1}{2}}-{e}^{-1}$$

$$=\dfrac{1}{\sqrt{e}}-\dfrac{1}{e}$$

$$=\dfrac{e-\sqrt{e}}{e\sqrt{e}}$$

$$=\dfrac{\sqrt{e}\left(\sqrt{e}-1\right)}{e\sqrt{e}}$$

$$=\dfrac{\sqrt{e}-1}{e}$$

View Full Answer

Its FREE, you're just one step away


Single Correct Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Single Correct Medium
Let $$f$$ be the continuous function on $$R$$ satisfying $$f\left ( x+y \right )= f\left ( x \right )+f\left ( y \right )$$ for all $$x,\: y\: \in \: R$$ with $$f\left ( 1 \right )= 2$$ and $$g$$ be a function satisfying $$f\left ( x \right )\div g\left ( x \right )= e^{x}$$ then the value of the integral $$\displaystyle \int_{0}^{1}f\left ( x \right )\: g\left ( x \right )\: dx$$ is
  • A. $$\dfrac{1}{e}-4$$
  • B. $$\displaystyle \frac{1}{4}\left ( e-2 \right )$$
  • C. $$\left (\dfrac {1}{2} \right )\left ( e-3 \right )$$
  • D. $$ \dfrac 2 3$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
$$\displaystyle \int \dfrac{t^2+t}{t} d t$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Solve:
$$\int {\frac{{dx}}{{x + {x^{ - 3}}}}} $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
$$\displaystyle \int_{0}^{1}\log \sin \left ( \frac{\pi }{2}x \right )dx=k \log \frac{1}{2}.$$ Find the value of $$k$$.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Let $$\displaystyle f\left ( x \right )=\frac{\sin 2x \cdot \sin \left ( \dfrac{\pi }{2}\cos x \right )}{2x-\pi }$$

Then answer the following question.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer