Mathematics

# Evaluate: $\displaystyle \int{e^x}\begin{pmatrix}\dfrac{x-1}{x^2}\end{pmatrix}dx$.

##### SOLUTION
$I=\int { { e }^{ x }\left( \dfrac { x-1 }{ { x }^{ 2 } } \right) }$

Let $\dfrac { 1 }{ x } =f(x),f'(x)=\dfrac { -1 }{ { x }^{ 2 } }$
$I=\int { { e }^{ x }\left( \frac { 1 }{ x } -\dfrac { 1 }{ { x }^{ 2 } } \right) }$

Using the property, we get:
${ e }^{ x }(f(x)+f'(x))dx={ e }^{ x }f(x)+c$
$I=\dfrac { { e }^{ x } }{ x } +c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 111

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle\int_{0}^{a}x^{4}\left ( a^{2}-x^{2} \right )^{1/2} dx$ equals
• A. $\displaystyle\frac{\pi a^{5}}{32}$
• B. $\displaystyle \frac{\pi a^{2}}{32}$
• C. None of these
• D. $\displaystyle\frac{\pi a^{6}}{32}$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int \frac{\sin x+\cos x}{\sqrt{\left ( 1+\sin 2x \right )}}$dx is
• A. $\displaystyle \sin x + C$
• B. $\displaystyle \cos x+C$
• C. $\displaystyle \tan x+C$
• D. $x+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Evaluate : $\int _ { \log 1 / 2 } ^ { \log 2 } \sin \left( \dfrac { e ^ { x } - 1 } { e ^ { x } + 1 } \right) d x$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Resolve into partial fraction $\displaystyle \frac{x^3-3x-2}{(x^2+x+1)(x+1)^2}$
• A. $\displaystyle \frac{3x-1}{x^2+x+1}+\frac{2}{(x+1)^2}+\frac{3}{(x+1)}$
• B. $\displaystyle \frac{3x}{x^2+x+1}+\frac{2}{(x+1)^2}-\frac{3}{(x+1)}$
• C. $\displaystyle \frac{x-1}{x^2+x+1}+\frac{2}{(x+1)^2}-\frac{3}{(x+1)}$
• D. $\displaystyle \frac{3x-1}{x^2+x+1}+\frac{2}{(x+1)^2}-\frac{3}{(x+1)}$

If $f,g,h$ be continuous functions on $[0,a]$ such that $f(a-x)=-f(x),g(a-x)=g(x)$ and $3h(x)-4h(a-x)=5$ then  $\displaystyle \int_0^a f(x)g(x)h(x)dx=0$