Question

How do you integrate $\int {{x^2}{e^{2x}}} $ by parts?

Answer 1

Hint: In this question we have the integration of two terms in multiplication therefore; we will use the formula of integration by parts and simplify it to get the final answer.

Formula used: $\int {uvdx} = u\int v dx - \int {\left( {\dfrac{{du}}{{dx}}\int {vdx} } \right)} $

Complete step-by-step solution:
We have the expression as:
$ \Rightarrow \int {{x^2}{e^{2x}}} dx$
Now it has $2$ terms which are ${x^2}$ and ${e^{2x}}$, in this question we will consider the first part to be $u$ and the latter part as $v$ and apply the formula of integration by parts.
Now on using the formula for integration by parts on the expression, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = {x^2}\int {{e^{2x}}} dx - \int {\left( {\dfrac{{d({x^2})}}{{dx}}\int {{e^{2x}}dx} } \right)} $
Now on integrating the first term, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = {x^2} \times \dfrac{{{e^{2x}}}}{2} - \int {\left( {\dfrac{{d({x^2})}}{{dx}}\int {{e^{2x}}dx} } \right)} $
Now on simplifying the first term, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \int {\left( {\dfrac{{d({x^2})}}{{dx}}\int {{e^{2x}}dx} } \right)} $
Now the first part is solved therefore, we will now proceed to the latter part of the expression. On taking the derivative, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \int {\left( {2x\int {{e^{2x}}dx} } \right)} $
Now on taking the integration, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \int {\left( {2x \times \dfrac{{{e^{2x}}}}{2}} \right)} $
Now on cancelling the terms and simplifying the expression, we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \int {\left( {\dfrac{{x{e^{2x}}}}{2}} \right)} $
Now we can take the constant value out of the integration and rewrite the expression as:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \dfrac{1}{2}\int {x{e^{2x}}} \to (1)$
Now the latter part cannot be integrated directly therefore, we will use the integration by parts formula on it by considering $u$ as $x$ and $v$ as ${e^{2x}}$.
We will solve the integration separately and substitute it in the equation. On using the formula on $\int {x{e^{2x}}} $we get:
$ \Rightarrow \int {x{e^{2x}}dx} = x\int {{e^{2x}}} dx - \int {\left( {\dfrac{{d(x)}}{{dx}}\int {{e^{2x}}dx} } \right)} $
On simplifying, we get:
$ \Rightarrow \int {x{e^{2x}}dx} = x \times \dfrac{{{e^{2x}}}}{2} - \int {\left( {1 \times \dfrac{{{e^{2x}}}}{2}} \right)} $
On simplifying, we get:
$ \Rightarrow \int {x{e^{2x}}dx} = \dfrac{{x{e^{2x}}}}{2} - \dfrac{1}{2}\int {{e^{2x}}} $
On integrating the term, we get:
$ \Rightarrow \int {x{e^{2x}}dx} = \dfrac{{x{e^{2x}}}}{2} - \dfrac{1}{2} \times \dfrac{{{e^{2x}}}}{2}$
On simplifying, we get:
$ \Rightarrow \int {x{e^{2x}}dx} = \dfrac{{x{e^{2x}}}}{2} - \dfrac{{{e^{2x}}}}{4}$
Now on substituting it back in equation $(1)$ we get:
$ \Rightarrow \int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \dfrac{{x{e^{2x}}}}{4} - \dfrac{{{e^{2x}}}}{8}$

$\int {{x^2}{e^{2x}}dx} = \dfrac{{{x^2}{e^{2x}}}}{2} - \dfrac{{x{e^{2x}}}}{4} - \dfrac{{{e^{2x}}}}{8}$ is the required solution.

Note: It is to be remembered that while doing integration by parts the terms $u$ and $v$ should follow the sequence of the acronym $LIATE$, which stands for logarithm, inverse, algebraic, trigonometric and exponential respectively.
It is to be remembered that integration and differentiation are inverse of each other. If the integration of $a$ is $b$ then the derivative of $b$ is $a$.