Mathematics

# If  $\int x ^ { 2 } \cdot e ^ { - 2 x } d x = e ^ { - 2 x } \left( a x ^ { 2 } + b x + c \right) + d,$  then

None of these

##### SOLUTION
$\int { { x }^{ 2 }{ e }^{ -2x }dx } ={ x }^{ 2 }\dfrac { { e }^{ -2x } }{ -2 } +\int { 2x.\dfrac { { e }^{ -2x } }{ -2 } }$
$=\dfrac { -{ x }^{ 2 } }{ 2 } { e }^{ -2x }-\int { x } { e }^{ -2x }$
$=\dfrac { { -x }^{ 2 } }{ 2 } { e }^{ -2x }-\left[ \dfrac { { xe }^{ -2x } }{ -2 } -\int { \dfrac { { e }^{ -2x } }{ -2 } } \right]$
$=\dfrac { { -x }^{ 2 } }{ 2 } { e }^{ -2x }-\left[ \dfrac { { -xe }^{ -2x } }{ 2 } +\dfrac { 1 }{ 2 } \dfrac { { e }^{ -2x } }{ -2 } \right]$
$=\dfrac { { -x }^{ 2 } }{ 2 } { e }^{ -2x }+\dfrac { { xe }^{ -2x } }{ 2 } +\dfrac { 1 }{ 4 } { e }^{ -2x }$
$={ e }^{ -2x }\left[ \dfrac { { -x }^{ 2 } }{ 2 } +\dfrac { x }{ 2 } +\dfrac { 1 }{ 4 } \right] +d$
$={ e }^{ -2x }\left[ { ax }^{ 2 }+bx+c \right] +d$
$a=-1/2\quad b=1/2\quad c=1/4$

Its FREE, you're just one step away

Single Correct Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109

#### Realted Questions

Q1 Subjective Medium
Solve:
$\displaystyle \int \cfrac{x^{2}+1}{1+x^{4}} d x$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
$\int {cosec\;x dx}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Express $\int_{0}^{2}e^{x} dx$ as the limit of a sum.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
$\displaystyle\int^{\pi}_{-\pi}\frac{\cos^2x}{1+a^x}dx=?$

$\int { \dfrac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x } } dx$