Solveeit Logo
Login

Question

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

Evaluate the following integral:
∫x2.e−2dx\int {{x^2}.} {e^{ - 2}}dx

Explanation

Solution

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

Complete step-by-step answer:
The given equation is ∫x2.e−2dx\int {{x^2}.} {e^{ - 2}}dx
In this question e−2{e^{ - 2}} is constant and we know the formula that,
∫mxndx=m∫xndx=mxn+1n+1+c \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.
=∫x2.e−2dx\int {{x^2}.} {e^{ - 2}}dx
Therefore,
=e−2∫x2dx =e−2x33+c =e−2(x33)+c  = {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(x33)+c   {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 ∫mxndx=m∫xndx=mxn+1n+1+c \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.