Question

Find the value of $\int {{2^x}{e^x}} dx$
$(A)\dfrac{{{2^x}{e^x}}}{{1 - \ln 2}} + C$
$(B)\dfrac{{{2^x}{e^x}}}{{1 + \ln 2}} + C$
$(C)\dfrac{{{2^x}{e^x}}}{{ - 1 + \ln 2}} + C$
$(D)\dfrac{{{2^x}{e^x}}}{{\ln (2e)}} + C$

Answer 1

Here, we will solve the given integral by using the formula of integration by parts and doing some calculation we will get the required answer.

Formula used: $\int {udv = uv} - \int {vdu}$

Complete step-by-step solution:
Let $I = \int {{2^x}{e^x}} dx$
We consider ${2^x}$ being $u$ and ${e^x}$ to be $v$ and use the integration by parts formula on $I$.
$I = {2^x}{e^x} - \int {{2^x}(\ln 2){e^x}dx}$
Since $\ln 2$ is a constant value we can take it outside the integral sign therefore, the equation can be re-written as:
$\Rightarrow$ $I = {2^x}{e^x} - \ln 2\int {{2^x}{e^x}dx}$
Since we know that $\int {{2^x}{e^x}} dx$ is equal to $I$, the equation can be written as:
$\Rightarrow$ $I = {2^x}{e^x} - (\ln 2)I$
On sending the similar term across the $=$ sign we get:
$\Rightarrow$ $I + (\ln 2)I = {2^x}{e^x}$
On taking $I$ common we get:
$\Rightarrow$ $(1 + \ln 2)I = {2^x}{e^x}$
Now sending the term across the $=$ sign, it will become the denominator, it can be written as:
$\Rightarrow$ $I = \dfrac{{{2^x}{e^x}}}{{1 + \ln 2}}$
Therefore,
$\Rightarrow$ $\int {{2^x}{e^x}} dx = \dfrac{{{2^x}{e^x}}}{{1 + \ln 2}} + c$

Therefore, the correct option is $(B)$

Note: The sum can be done using another method which is the following:
$I = \int {{2^x}{e^x}} dx$
Now using the rule of exponent, we can rewrite the equation as:
$\Rightarrow$ $I = \int {{{(2e)}^x}} dx$
Here we have to formula we get,
$\Rightarrow$ $\int {{a^x} = \dfrac{{{a^x}}}{{\log a}}} + c$
Since the value $2e$ is a constant value therefore $a$ can be considered as $2e$.
On integrating we get:
$\Rightarrow$ $I = \dfrac{{{{(2e)}^x}}}{{\ln (2e)}} + c$
Now the denominator can be re-split into the original form using the rule of exponents, it can be written as:
$\Rightarrow$ $I = \dfrac{{{2^x}{e^x}}}{{\ln (2e)}} + c$, which is the required answer.
Therefore, the correct option using this method is option $(D)$.
An answer in integration is never strictly a single answer; the answer could vary based on the method used to solve it.
All the basic integration and derivative formulas should be memorized by a student to solve these types of questions.
Integration and derivation are opposites of each other example if integration of a term $A$ is $B$, then the derivative of term $B$ will be $A$.