Question

Evaluate the following integral:
$\int {{x^2}.} {e^{ - 2}}dx$

Answer 1

Hint: To solve such types of questions can simply be solved using the basic formulae of integration. As ${e}^{-2}$ is constant. We need to solve only the term containing x.

Complete step-by-step answer:
The given equation is $\int {{x^2}.} {e^{ - 2}}dx$
In this question ${e^{ - 2}}$ is constant and we know the formula that,
$
  \int {m{x^n}} dx
   = m{\int x ^n}dx
   = m\dfrac{{{x^{n + 1}}}}{{n + 1}} + c $
  where m is constant and c is constant of integration.
=$\int {{x^2}.} {e^{ - 2}}dx$
Therefore,
$
   = {e^{ - 2}}\int {x^2}dx \\
   = {e^{ - 2}}\{ \dfrac{{{x^3}}}{3}\} + c \\
   = {e^{ - 2}}(\dfrac{{{x^3}}}{3}) + c \\
 $
Hence, the answer to this question is $
  {e^{ - 2}}(\dfrac{{{x^3}}}{3}) + c \\
    \\
 $ where c is constant of integration.

Note: For these type of questions we must remember and practice the basic formulae of integration as $
  \int {m{x^n}} dx
    = m{\int x ^n}dx
   = m\dfrac{{{x^{n + 1}}}}{{n + 1}} + c
 $ where m is constant and c is constant of integration. Doing this will solve your problem.