Mathematics

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


ANSWER

None of these


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

Its FREE, you're just one step away


Single Correct Hard Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 109
Enroll Now For FREE

Realted Questions

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

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

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Subjective Hard
$$\int { \dfrac { { sec }^{ 2 }x }{ { cosec }^{ 2 }x }  } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer