Mathematics

# If $\displaystyle \int {\left (\dfrac { x-1 }{ { x }^{ 2 } } \right ){ e }^{ x }dx=f\left( x \right) { e }^{ x }+c }$, then write the value of f(x).

##### SOLUTION
$\displaystyle\int \dfrac{x-1}{x^2}\times e^{x} d x=\int e^{x}\bigg(\dfrac{1}{x}-\dfrac{1}{x^2}\bigg) dx$

This is in the form of $\displaystyle\int e^{x}(g(x)+g'(x)) d x=e^{x}g(x)+C$

Here $g(x)=\dfrac{1}{x}$ and $g'(x)=-\dfrac{1}{x^2}$

$\implies \displaystyle\int (\dfrac{x-1}{x^2})e^{x} d x=\dfrac{e^{x}}{x}+C$

$\implies f(x)=\dfrac{1}{x}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int\frac{x+\sin x}{1+\cos x}dx=$
• A. $xcot \displaystyle \frac{x}{2}+c$
• B. $xsin \displaystyle \frac{x}{2}+c$
• C. $xcos \displaystyle \frac{x}{2}+c$
• D. $xtan \displaystyle \frac{x}{2}+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\int {{e^{3{{\log }_e}x}}.{{\left( {{x^4} + 1} \right)}^{ - 1}}dx = \_\_\_\_\_\_\_\_\_ + C.}$
• A. $\log \left( {{x^4} + 1} \right)$
• B. $-\log \left( {{x^4} + 1} \right)$
• C. $\frac{{ - 3}}{{{{\left( {{x^4} + 1} \right)}^3}}}$
• D. $\frac{1}{4}\log \left( {{x^4} + 1} \right)$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
The value of $\displaystyle\int { \cfrac { \sin { x } +\cos { x } }{ 3+\sin { 2x } } } dx$ is
• A. $\cfrac { 1 }{ 2 } \log { \left( \cfrac { 2+\sin { x } }{ 2-\sin { x } } \right) +C }$
• B. $\cfrac { 1 }{ 4 } \log { \left( \cfrac { 1+\sin { x } }{ 1-\sin { x } } \right) +C }$
• C. None of the above
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1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Hard
Evaluate: $\int ^{1}_{0}x (\tan^{-1}x)^{2}dx$

$\int_{}^{} {\frac{{ - 1}}{{\sqrt {1 - {x^2}} }}dx}$